\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc784\uc758\uc758 \\(x\\in[a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) > 0\\)\uc77c \ub54c, \\(x\\)\ucd95\uacfc \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504, \uadf8\ub9ac\uace0 \ub450 \uc9c1\uc120 \\(x=a,\\) \\(x=b\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \ubd80\ubd84\uc758 \ub113\uc774\ub97c \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc815\uc801\ubd84\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\uc640 \uac19\uc740 \uc815\uc758\ub294 \uc9c1\uad00\uc801\uc778 \uc815\uc758\uc774\uba70 \uc5f0\uc18d\ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub9cc \uc815\uc758\ub418\ubbc0\ub85c \ub300\ub2e8\ud788 \ud611\uc18c\ud558\ub2e4.<\/p>\n
\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ub9ac\ub9cc \uc801\ubd84\uc744 \uc5c4\ubc00\ud558\uac8c \uc815\uc758\ud558\uace0, \uc801\ubd84 \uac00\ub2a5\uc131\uacfc \uc815\uc801\ubd84\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n
<\/p>\n
\\(I = [a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[x_i = a + \\frac{b-a}{n} i \\,\\,\\, (i = 0,\\,1,\\,2,\\,\\cdots,\\,n) \\]
\n\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \uc810 \\(x_i\\)\ub4e4\uc5d0 \uc758\ud558\uc5ec \uad6c\uac04 \\([a,\\,b]\\)\ub294 \uae38\uc774\uac00 \uac19\uc740 \\(n\\)\uac1c\uc758 \uc18c\uad6c\uac04
\n\\[[x_0 ,\\, x_1 ] ,\\, [x_1 ,\\, x_2] ,\\, \\cdots ,\\, [x_{n-1} ,\\, x_n ]\\]
\n\uc73c\ub85c \ucabc\uac1c\uc5b4 \uc9c4\ub2e4. \\(i\\)\ubc88\uc9f8 \uc18c\uad6c\uac04\uc758 \uae38\uc774\ub97c \\(\\Delta x_i\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(i\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\Delta x_i = x_i – x_{i-1} = \\frac{b-a}{n} \\quad (i=1,\\,2,\\,\\cdots,\\,n)\\]
\n\uc774\ub2e4. \ub9cc\uc57d \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ud568\uc22b\uac12\uc774 \ubaa8\ub450 \uc591\uc218\ub77c\uba74, \\(f \\left( x_i \\right) \\Delta x_i\\)\ub294 \uac00\ub85c\uc758 \uae38\uc774\uac00 \\(\\Delta x_i\\)\uc774\uace0 \uc138\ub85c\uc758 \uae38\uc774\uac00 \\(f \\left( x_i \\right)\\)\uc778 \uc9c1\uc0ac\uac01\ud615 \uc870\uac01\uc758 \ub113\uc774\uac00 \ub41c\ub2e4.<\/p>\n
\ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0, \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc640 \\(x\\)\ucd95, \uadf8\ub9ac\uace0 \ub450 \uc9c1\uc120 \\(x=a,\\) \\(x=b\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \ubd80\ubd84\uc758 \ub113\uc774\ub97c \\(A\\)\ub77c\uace0 \ud558\uc790. \uc9c1\uc0ac\uac01\ud615 \uc870\uac01\ub4e4\uc758 \ub113\uc774\uc758 \ud569 \uad6c\ubd84\uad6c\uc801\ubc95\uc73c\ub85c \uc815\uc801\ubd84\uc744 \uc815\uc758\ud558\uba74 \uc5f0\uc18d\uc774 \uc544\ub2cc \ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ub17c\ub9ac\uc801\uc73c\ub85c \ud310\ubcc4\ud558\uae30 \uc5b4\ub835\ub2e4\ub294 \ub2e8\uc810\uc774 \uc788\ub2e4. \uc608\ucee8\ub300 \uadf8\ub7ec\ubbc0\ub85c \ub354 \uc77c\ubc18\uc801\uc778 \ud568\uc218\uc758 \uc801\ubd84\uc744 \ub2e4\ub8f0 \uc218 \uc788\ub294 \uc815\uc758\ub97c \ub3c4\uc785\ud574\uc57c \ud55c\ub2e4.<\/p>\n <\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc810 \\(x_i\\)\ub4e4\uc774 \ub9cc\uc57d \\(a\\le x \\le b\\)\uc758 \ubc94\uc704\uc5d0\uc11c \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc640 \\(x\\)\ucd95 \uc0ac\uc774\uc758 \ub113\uc774\uac00 \uc874\uc7ac\ud558\uace0 \uadf8 \uac12\uc774 \\(A\\)\ub77c\uba74, \ubd84\ud560 \\(P\\)\uc5d0 \uc758\ud55c \uc18c\uad6c\uac04\uc758 \uae38\uc774\uac00 \uc791\uc744\uc218\ub85d \ub9ac\ub9cc\ud569 (3)\uc758 \uac12\uc740 \\(A\\)\uc5d0 \uac00\uae4c\uc6cc\uc9c8 \uac83\uc774\ub2e4. \uadf8\ub7ec\ub098 \ud55c \uad6c\uac04 \ub0b4\uc5d0\uc11c \uc18c\uad6c\uac04\uc758 \uae38\uc774\ub294 \ud558\ub098\uac00 \uc544\ub2c8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubd84\ud560\uc758 \uc18c\uad6c\uac04\uc758 \uae38\uc774\ub97c \ub300\ud45c\ud560 \uc218 \uc788\ub294 \uac12\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n \\(P = \\left\\{ x_0,\\, x_1 ,\\, \\cdots ,\\, x_n \\right\\}\\)\uc774 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc77c \ub54c \\(P\\)\uc5d0 \uc758\ud574 \ub9cc\ub4e4\uc5b4\uc9c4 \uc18c\uad6c\uac04\uc758 \uae38\uc774 \uc911 \uac00\uc7a5 \ud070 \uac12\uc744 \\(P\\)\uc758 \ub178\ub984<\/span>(norm) \ub610\ub294 \uae30\uc900\uae38\uc774<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \\(\\lVert P \\rVert\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \uc801\ubd84\uc744 \uc815\ud655\ud558\uac8c \uc815\uc758\ud558\uae30 \uc704\ud558\uc5ec \uba87 \uac00\uc9c0 \uc6a9\uc5b4\uc640 \uae30\ud638\ub97c \ub354 \uc815\uc758\ud558\uc790. \uad6c\uac04 \\([a,\\,b]\\)\uc758 \ubaa8\ub4e0 \ubd84\ud560\uc758 \ubaa8\uc784\uc744 \\(\\mathcal{P}([a,\\,b])\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub9ac\uace0 \ubd84\ud560 \\(P\\)\uc5d0 \uc758\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c4 \uc18c\uad6c\uac04\uc5d0\uc11c \ud55c \uc810\uc529 \ud0dd\ud558\uc5ec \ub9cc\ub4e0 \ubaa8\ub4e0 \uc720\ud55c\uc218\uc5f4\uc758 \ubaa8\uc784\uc744 \\(\\mathcal{S}(P)\\)\ub85c \ub098\ud0c0\ub0b4\uc790. [\uc774\ub7ec\ud55c \ud45c\uae30\ubc95\uc740 \ucc45\ub9c8\ub2e4 \ub2e4\ub974\ub2e4.]<\/p>\n \uad6c\uac04 \\([a,\\,b]\\)\uc758 \ubd84\ud560 \\(P\\)\uc758 \ub178\ub984 \\(\\lVert P \\rVert\\)\uac00 \uc791\uc544\uc9c0\uba74 \\(P\\)\uc5d0 \uc758\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c4 \ubaa8\ub4e0 \uc18c\uad6c\uac04\uc758 \uae38\uc774\uac00 \uc791\uc544\uc9c4\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud55c \uc18c\uad6c\uac04\uc5d0\uc11c \ud0dd\ud560 \uc218 \uc788\ub294 \uc810\uc774 \ub193\uc77c \uc218 \uc788\ub294 \ubc94\uc704 \ub610\ud55c \uc791\uc544\uc9c4\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\lVert P \\rVert\\)\uac00 \\(0\\)\uc5d0 \ud55c \uc5c6\uc774 \uac00\uae4c\uc6cc\uc9c8 \ub54c \ud45c\uc9d1\uc218\uc5f4\uc744 \uc5b4\ub290\uac83\uc744 \ud0dd\ud558\ub4e0 \uc0c1\uad00 \uc5c6\uc774 \ub9ac\ub9cc \ud569 \\(S(f,\\,P,\\,\\xi)\\)\uac00 \uc77c\uc815\ud55c \uac12\uc5d0 \uac00\uae4c\uc6cc\uc9c0\ub294 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n \uc815\uc758 1. (\ub9ac\ub9cc \ud569\uc758 \uadf9\ud55c)<\/span><\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc774\uba70 \\(J\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\lVert P \\rVert < \\delta\\)\uc778 \ubaa8\ub4e0 \ubd84\ud560 \\(P \\in \\mathcal{P}([a,\\,b])\\)\uc640 \ubaa8\ub4e0 \ud45c\uc9d1\uc218\uc5f4 \\(\\xi \\in \\mathcal{S}(P)\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\lvert S(f,\\,P,\\,\\xi ) - J \\rvert < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud558\uba74 \u2018\\(\\lVert P \\rVert \\to 0\\)\uc77c \ub54c \\(S(f,\\,P,\\,\\xi )\\)\ub294 \\(J\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uae30\ud638\ub85c\ub294\n\\[\\lim_{\\lVert P \\rVert \\to 0} S(f,\\,P,\\,\\xi ) = J\\tag{4}\\]\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc5ec\uae30\uc11c \ud45c\uc9d1\uc218\uc5f4 \\(\\xi\\)\ub294 \uadf9\ud55c\uac12 \\(J\\)\uc5d0 \uc601\ud5a5\uc744 \ub07c\uce58\uc9c0 \uc54a\uc73c\ubbc0\ub85c, (4)\ub97c \uac04\ub2e8\ud788\n\\[\\lim_{\\lVert P \\rVert \\to 0} S(f,\\,P ) = J\\tag{5}\\]\n\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud558\uba70, \n\\[\\lim_{\\lVert P \\rVert \\to 0} \\sum_{P} f\\left(\\xi_i \\right) \\Delta x_i =J\\tag{6}\\]\n\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.\n<\/p>\n<\/div>\n \uc774\uc81c \ube44\ub85c\uc18c \ub9ac\ub9cc \uc801\ubd84\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n \uc815\uc758 2. (\ub9ac\ub9cc \uc801\ubd84)<\/span><\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc2e4\uc218 \\(J\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc774\uc640 \uac19\uc774 \uc815\uc758\ub41c \uc801\ubd84\uc740 \uad6c\ubd84\uad6c\uc801\ubc95\uc5d0 \ube44\ud558\uc5ec \ub354 \ub113\uc740 \ubc94\uc704\uc758 \ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud560 \uc218 \uc788\uac8c \ud574\uc900\ub2e4.<\/p>\n \uc608\uc81c 1.<\/span> \ud480\uc774.<\/span> \ub9cc\uc57d \\(J > 0\\)\uc774\ub77c\uba74 \\(\\epsilon = J\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\delta > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(n > 3\/\\delta\\)\uc778 \uc790\uc5f0\uc218 \\(n\\)\uc744 \ud0dd\ud558\uace0 \uc774\ubc88\uc5d0\ub294 \\(J \\le 0\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(\\epsilon = 1\\)\uc774\ub77c\uace0 \ud558\uace0 \\(\\delta > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(n > 3\/\\delta\\)\uc778 \uc790\uc5f0\uc218 \\(n\\)\uc744 \ud0dd\ud558\uace0 \uc55e\uc5d0\uc11c\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \ubd84\ud560 \\(P\\)\ub97c \uc7a1\uc740 \ud6c4 \uac01 \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\uc5d0\uc11c \uc720\ub9ac\uc218 \\(\\xi_i\\)\ub97c \ud558\ub098\uc529 \ud0dd\ud558\uc5ec \uc218\uc5f4 \\(\\xi = \\left\\{ \\xi_i \\right\\}\\)\ub97c \ub9cc\ub4e4\uc790. \uadf8\ub7ec\uba74 \uc694\ucee8\ub300 \uc5b4\ub5a0\ud55c \uc2e4\uc218\ub77c\ub3c4 \\(f\\)\uc758 \ub9ac\ub9cc \uc801\ubd84\uac12\uc774 \ub420 \uc218 \uc5c6\uc73c\ubbc0\ub85c \\(f\\)\ub294 \\([2,\\,5]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4. \ud558\uc9c0\ub9cc \uc815\uc758 2\ub9cc \uc0ac\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud558\ub294 \uac83\uc740 \uc0c1\ub2f9\ud788 \ubcf5\uc7a1\ud558\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc815\uc758 2\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud558\uae30 \uc704\ud574\uc11c\ub294 \uc801\ubd84\uac12 \\(J\\)\uac00 \uc788\uc5b4\uc57c \ud558\uae30 \ub54c\ubb38\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ub354 \uc27d\uac8c \ud310\ubcc4\ud560 \uc218 \uc788\ub294 \ubc29\ubc95\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n <\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \u2018\uc720\uacc4\uc778\u2019 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\[P = \\left\\{ x_0 ,\\, x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_n \\right\\}\\]\uc774 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\ub77c\uace0 \ud558\uc790. \uac01 \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\ud55c\uc744 \\(M_i (f,\\,P)\\)\ub85c \ud45c\uae30\ud558\uace0, \\(f\\)\uc758 \ud558\ud55c\uc744 \\(m_i (f,\\,P)\\)\ub85c \ud45c\uae30\ud55c\ub2e4. \uc989 \\(P_1\\)\uacfc \\(P_2\\)\uac00 \ubaa8\ub450 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\uace0 \\(P_1 \\subseteq P_2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(P_2\\)\uc5d0 \uc758\ud574 \ub9cc\ub4e4\uc5b4\uc9c4 \uc18c\uad6c\uac04\uc758 \uae38\uc774\ub294 \\(P_1\\)\uc5d0 \uc758\ud574 \ub9cc\ub4e4\uc5b4\uc9c4 \uc18c\uad6c\uac04\uc758 \uae38\uc774 \uc774\ud558\uc774\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \\(P_1 \\subseteq P_2\\)\uc77c \ub54c \\(P_2\\)\ub97c \\(P_1\\)\uc758 \uc138\ub828\ubd84\ud560<\/span>(refinement)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. <\/p>\n \ud568\uc218 \\(f\\)\uac00 \\([x_{i-1} ,\\, x_i ]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(x_{i-1} < y_i < x_i \\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74\n\\[\\sup_{x_{i-1} \\le x \\le x_i} f(x) \\cdot (x_i - x_{i-1}) \\ge \\sup_{x_{i-1} \\le x \\le y_i} f(x) \\cdot (y_i - x_{i-1}) + \\sup_{y_i \\le x \\le x_i} f(x) \\cdot (x_i - y_i)\\]\n\uc774\ubbc0\ub85c, \\(P_1 \\subseteq P_2\\)\uc77c \ub54c \\(U(f,\\,P_1 ) \\ge U(f,\\,P_2 )\\)\uc774\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c\n\\[\\inf_{x_{i-1} \\le x \\le x_i} f(x) \\cdot (x_i - x_{i-1}) \\le \\inf_{x_{i-1} \\le x \\le y_i} f(x) \\cdot (y_i - x_{i-1}) + \\inf_{y_i \\le x \\le x_i} f(x) \\cdot (x_i - y_i)\\]\n\uc774\ubbc0\ub85c, \\(P_1 \\subseteq P_2\\)\uc77c \ub54c \\(L(f,\\,P_1 ) \\le L(f,\\,P_2 )\\)\uc774\ub2e4. \uc694\ucee8\ub300 \ubd84\ud560\uc744 \uc138\ub828\ud560 \uc218\ub85d \uc0c1\ud569\uc740 \uc791\uc544\uc9c0\uace0 \ud558\ud569\uc740 \ucee4\uc9c4\ub2e4.<\/p>\n \\(P_1\\)\uacfc \\(P_2\\)\uac00 \ubaa8\ub450 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\uace0 \\(P_3 \\supseteq P_1 \\cup P_2\\)\uc774\uba74 \\(P_3\\)\uc740 \\(P_1\\)\uc758 \uc138\ub828\ubd84\ud560\uc774\uba74\uc11c \\(P_2\\)\uc758 \uc138\ub828\ubd84\ud560\uc774\ub2e4. \uc774\ub7ec\ud55c \ubd84\ud560 \\(P_3\\)\uc744 \\(P_1\\)\uacfc \\(P_2\\)\uc758 \uacf5\ud1b5\uc138\ub828\ubd84\ud560<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\ub54c \uc815\uc758 3. (\ub2e4\ub974\ubd80 \uc801\ubd84)<\/span><\/p>\n \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0, \uc774 \uad6c\uac04\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc774 \uac19\uc73c\uba74, \uc989 \ub2e4\ub974\ubd80 \uc801\ubd84\uacfc \ub9ac\ub9cc \uc801\ubd84\uc744 \uac19\uc740 \uae30\ud638\ub85c \ub098\ud0c0\ub0b4\ub294 \uc774\uc720\ub294 \uc0ac\uc2e4 \uc720\uacc4\uc778 \ud568\uc218\uc5d0 \ub300\ud558\uc5ec \ub2e4\ub974\ubd80 \uc801\ubd84\uacfc \ub9ac\ub9cc \uc801\ubd84\uc774 \ub3d9\uce58\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. <\/p>\n \uc815\ub9ac 1. (\ub9ac\ub9cc \ud310\uc815\ubc95)<\/span><\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uc73c\uba70 \uc774 \uad6c\uac04\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc99d\uba85<\/p>\n \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uace0 \uc801\ubd84\uac12\uc744 \\(J\\)\ub77c\uace0 \ud558\uc790. \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc815\uc758 3\uc5d0 \uc758\ud558\uc5ec \uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \uc989 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec (8)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \ub9ac\ub9cc \ud310\uc815\ubc95\uc758 \uc7a5\uc810\uc740 \uc801\ubd84\uac12 \\(J\\)\ub97c \uc54c\uc9c0 \ubabb\ud558\ub354\ub77c\ub3c4 \uc8fc\uc5b4\uc9c4 \ud568\uc218\uac00 \uc801\ubd84 \uac00\ub2a5\ud55c\uc9c0 \ud310\ubcc4\ud560 \uc218 \uc788\ub2e4\ub294 \uac83\uc774\ub2e4.<\/p>\n \uc608\uc81c 2.<\/span> \ud480\uc774.<\/span> \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\([a,\\,b]\\)\uc758 \ubd84\ud560 \uc911\uc5d0\uc11c \\(\\lVert P \\rVert < \\epsilon \/(b-a)\\)\uc778 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud558\uc790. \uadf8\ub7ec\uba74 \uac01 \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\uc5d0\uc11c\n\\[\\begin{gather}\nM_i = f(x_i ) ,\\,\\, m_i = f(x_{i-1} ) ,\\\\[8pt] f(x_0) = f(a) ,\\,\\, f(x_n) = f(b)\n\\end{gather}\\]\n\uc774\ubbc0\ub85c\n\\[\\begin{align}\nU (f,\\,P) - L(f,\\,P) &= \\sum_{i=1}^n (M_i - m_i ) \\Delta x_i \\\\[6pt]\n&= \\sum_{i=1}^n (f(x_i ) - f(x_{i-1} ))\\Delta x_i \\\\[6pt]\n&\\le \\sum_{i=1}^n (f(x_i ) - f(x_{i-1} )) \\lVert P \\rVert \\\\[6pt]\n&= (f(b) - f(a)) \\lVert P \\rVert < \\epsilon\n\\end{align}\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.\n<\/span><\/p>\n<\/div>\n <\/p>\n \uc774\uc81c \ub9ac\ub9cc \uc801\ubd84\uacfc \ub2e4\ub974\ubd80 \uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uba3c\uc800 \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc744 \uc0b4\ud3b4\ubcf4\uace0, \uadf8\uac83\uc774 \uacb0\uad6d \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uacfc \ud544\uc694\ucda9\ubd84\uc870\uac74\uc784\uc744 \ubc1d\ud790 \uac83\uc774\ub2e4.<\/p>\n \uba3c\uc800 \ub9ac\ub9cc \ud569\uc744 \uc774\uc6a9\ud558\uc5ec \ub2e4\ub974\ubd80 \uc801\ubd84\uacfc \ub3d9\uce58\uc778 \uc801\ubd84\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n \uc815\ub9ac 2. (\uc138\ub828\ubd84\ud560\uc744 \uc774\uc6a9\ud55c \ub9ac\ub9cc \ud569\uc758 \uadf9\ud55c)<\/span><\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uc73c\uba70 \uc774 \uad6c\uac04\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc2e4\uc218 \\(J\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \ubd84\ud560 \\(P_\\epsilon\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(P\\supseteq P_\\epsilon\\)\uc778 \uc784\uc758\uc758 \ubd84\ud560 \\(P\\)\uc640 \ubaa8\ub4e0 \ud45c\uc9d1\uc218\uc5f4 \\(\\xi \\in \\mathcal{S}(P)\\)\uc5d0 \ub300\ud558\uc5ec \uc99d\uba85<\/p>\n \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uace0 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ub2e4\ub974\ubd80 \uc801\ubd84\uac12\uc744 \\(J\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\([a,\\,b]\\)\uc758 \ubd84\ud560 \\(P_\\epsilon\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\([a,\\,b]\\)\uc758 \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(\\xi \\in \\mathcal{S}(P)\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc815\ub9ac\ub294 \ub9ac\ub9cc \uc801\ubd84\uacfc \ub2e4\ub974\ubd80 \uc801\ubd84\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud55c\ub2e4.<\/p>\n \uc815\ub9ac 3. (\ub9ac\ub9cc \uc801\ubd84\uacfc \ub2e4\ub974\ubd80 \uc801\ubd84\uc758 \uad00\uacc4)<\/span><\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uc73c\uba70 \uc774 \uad6c\uac04\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud55c \uac83\uc774\ub2e4. \uc774\ub54c \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ub9ac\ub9cc \uc801\ubd84\uac12\uacfc \ub2e4\ub974\ubd80 \uc801\ubd84\uac12\uc740 \uc77c\uce58\ud55c\ub2e4.<\/p>\n<\/div>\n \uc99d\uba85<\/p>\n \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \\(f\\)\uc758 \ub9ac\ub9cc \uc801\ubd84\uac12\uc774 \\(J\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc815\uc758 1\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(\\lVert P_\\epsilon \\rVert < \\delta\\)\uc778 \\([a,\\,b]\\)\uc758 \ubd84\ud560 \\(P_\\epsilon\\)\uc744 \ud0dd\ud558\uc790. \uadf8\ub9ac\uace0 \\(P\\)\uac00 \\(P_\\epsilon\\)\uc758 \uc138\ub828\ubd84\ud560\uc774\ub77c\uace0 \ud558\uc790. \uc138\ub828\ubd84\ud560\uc758 \ub178\ub984\uc740 \ubcf8\ub798\uc758 \ubd84\ud560\uc758 \ub178\ub984 \uc774\ud558\uc774\ubbc0\ub85c \\(\\lVert P \\rVert < \\delta\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(f\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \uc784\uc758\uc758 \\(\\xi \\in \\mathcal{S}(P)\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert S(f,\\,P,\\,\\xi ) - J \\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989 \\(P_\\epsilon\\)\uc758 \uc784\uc758\uc758 \uc138\ub828\ubd84\ud560 \\(P\\)\uc640 \ud45c\uc9d1\uc218\uc5f4 \\(\\xi\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert S(f,\\,P,\\,\\xi ) - J \\rvert < \\epsilon\\)\uc774\ubbc0\ub85c \uc815\ub9ac 2\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n \uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \\(f\\)\uc758 \ub2e4\ub974\ubd80 \uc801\ubd84\uac12\uc774 \\(J\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\([a,\\,b]\\)\uc758 \ubd84\ud560 \uc774\uc81c \\(P = \\left\\{ x_0 ,\\, x_1 ,\\, \\cdots ,\\, x_n \\right\\}\\)\uc774 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\uace0 \\(\\lVert P \\rVert < \\delta\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\xi \\in \\mathcal{S}(P)\\)\ub77c\uace0 \ud558\uc790. <\/p>\n \\(P\\)\uc5d0 \uc758\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c4 \ud55c \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<\/p>\n \ub9cc\uc57d \\([x_{i-1} ,\\, x_i ] \\subseteq [y_{j-1} ,\\, y_j ]\\)\ub77c\uba74, \uc989 \\([x_{i-1} ,\\, x_i]\\)\uac00 \\(P_1\\)\uc758 \ud55c \uc18c\uad6c\uac04\uc5d0 \ud3ec\ud568\ub41c\ub2e4\uba74, \ub9cc\uc57d \\([x_{i-1} ,\\, x_i]\\)\uac00 \\(P_1\\)\uc758 \ud55c \uc18c\uad6c\uac04\uc5d0 \ud3ec\ud568\ub418\uc9c0 \uc54a\ub294\ub2e4\uba74 \\([x_{i-1} ,\\, x_i]\\)\ub294 \\(P_1\\)\uc758 \ub450 \uac1c\uc758 \uc18c\uad6c\uac04\uc5d0 \ud3ec\ud568\ub41c\ub2e4. \\(\\delta\\)\uac00 \ucda9\ubd84\ud788 \uc791\uc73c\ubbc0\ub85c \\([x_{i-1} ,\\, x_i]\\)\uac00 \\(P_1\\)\uc758 \uc138 \uac1c\uc758 \uc18c\uad6c\uac04\uc5d0 \uac78\uccd0\uc788\uc9c0\ub294 \uc54a\ub294\ub2e4. \uc989 \uc774\ub85c\uc368 \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc77c \ub54c \ub2e4\uc74c \ub124 \uc9c4\uc220\uc740 \ubaa8\ub450 \ub3d9\uce58\uc774\ub2e4.<\/p>\n \ub9cc\uc57d \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uba74 \uc5b4\ub5a0\ud560\uae4c? \uba3c\uc800 \ub2e4\ub974\ubd80 \uc801\ubd84\uc740 \\(f\\)\uac00 \uc720\uacc4\uc778 \uacbd\uc6b0\uc5d0\ub9cc \uc815\uc758\ud588\uc73c\ubbc0\ub85c, \\(f\\)\uac00 \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0\uc5d0 \ub2e4\ub974\ubd80 \uc801\ubd84\uc740 \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4. \ud55c\ud3b8 \\(f\\)\uac00 \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n \uc815\ub9ac 4. (\uc720\uacc4\uac00 \uc544\ub2cc \ud568\uc218\uc758 \ub9ac\ub9cc \uc801\ubd84)<\/span><\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n \uc99d\uba85<\/p>\n \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. [\uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \\(f\\)\uac00 \uc774 \uad6c\uac04\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\uc774\uace0 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uba74 \\(-f\\)\ub294 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2cc \ud568\uc218\uac00 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.]<\/p>\n \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ubbc0\ub85c \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(f(c_n ) > n\\)\uc778 \uc810 \\(c_n\\)\uc774 \\((a,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \\(\\left\\{ c_n \\right\\}\\)\uc740 \uc720\uacc4\uc778 \uc218\uc5f4\uc774\ubbc0\ub85c \ubcfc\ucc28\ub178-\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4 \\(\\left\\{ c_{n_k} \\right\\}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774 \ubd80\ubd84\uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \\(c\\)\ub77c\uace0 \ud558\uc790.<\/p>\n \uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \ub9ac\ub9cc \uc801\ubd84\uac12\uc744 \\(J\\)\ub77c\uace0 \ud558\uc790. \uc774\uc81c \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uac12\uc774 \\(J\\)\uac00 \ub420 \uc218 \uc5c6\uc74c\uc744 \ubcf4\uc77c \uac83\uc774\ub2e4. \\(\\epsilon = 1\\)\uc774\ub77c\uace0 \ud558\uace0 \\(\\delta > 0\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(i \\ne j\\)\uc778 \uacbd\uc6b0, \uac01 \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\uc5d0\uc11c \uc810 \\(\\xi_i\\)\ub97c \uc544\ubb34\uac70\ub098 \ud558\ub098\uc529 \ud0dd\ud55c\ub2e4. \uadf8\ub7ec\uba74 \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0 \\(f\\)\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc740 \ub9ac\ub9cc \uc801\ubd84\uc5d0\uc11c\ub4e0 \ub2e4\ub974\ubd80 \uc801\ubd84\uc5d0\uc11c\ub4e0 \ub2e4\ub8f0 \ud544\uc694\uac00 \uc5c6\ub2e4. \ub530\ub77c\uc11c \uc9c0\uae08\ubd80\ud130\ub294 \ub9ac\ub9cc \uc801\ubd84\uacfc \ub2e4\ub974\ubd80 \uc801\ubd84\uc744 \uad6c\ubd84\ud558\uc9c0 \uc54a\uace0 \ubaa8\ub450 \ub9ac\ub9cc \uc801\ubd84\uc774\ub77c\uace0 \ubd80\ub974\uc790. \ub9cc\uc138!<\/p>\n <\/p>\n \uc774\uc81c \uc6b0\ub9ac\ub294 \uc8fc\uc5b4\uc9c4 \ud568\uc218\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud55c\uc9c0 \ud310\ubcc4\ud560 \ub54c \ub9ac\ub9cc \uc801\ubd84\uc758 \uc815\uc758, \ub2e4\ub974\ubd80 \uc801\ubd84\uc758 \uc815\uc758, \ub9ac\ub9cc \ud310\uc815\ubc95 \uc911 \uc6d0\ud558\ub294 \uac83\uc744 \uc790\uc720\ub86d\uac8c \ud0dd\ud558\uc5ec \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n \uc815\ub9ac 5. (\uc5f0\uc18d\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131)<\/span><\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n \uc99d\uba85<\/p>\n \ub9ac\ub9cc \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uc790. \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\([a,\\,b]\\)\uac00 \ub2eb\ud78c \uad6c\uac04\uc774\ubbc0\ub85c \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\([a,\\,b]\\)\uc5d0 \uc18d\ud558\ub294 \uc784\uc758\uc758 \uc810 \\(s\\)\uc640 \\(t\\)\uc5d0 \ub300\ud558\uc5ec \uad6c\ubd84\uad6c\uc801\ubc95\uc73c\ub85c \ubaa8\ub4e0 \ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud560 \uc218\ub294 \uc5c6\uc9c0\ub9cc, \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uc774 \ubcf4\uc7a5\ub41c \ud568\uc218\uc758 \uacbd\uc6b0\uc5d0\ub294 \uad6c\ubd84\uad6c\uc801\ubc95\uc73c\ub85c \uc801\ubd84\uac12\uc744 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\n \uc815\ub9ac 6. (\uad6c\ubd84\uad6c\uc801\ubc95\uc744 \uc774\uc6a9\ud55c \uc801\ubd84\uac12\uc758 \uacc4\uc0b0)<\/span><\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uba74 \uc801\ubd84\uac12\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \uc99d\uba85<\/p>\n \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ub9ac\ub9cc \uc801\ubd84\uac12\uc744 \\(J\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\([a,\\,b]\\)\ub97c \\(n\\)\ub4f1\ubd84\ud55c \ubd84\ud560\uc744 \\(P_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uc989 \uc774\uc81c \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\lVert P \\rVert < \\delta\\)\uc778 \ubd84\ud560 \\(P\\)\uc640 \uc784\uc758\uc758 \ud45c\uc9d1\uc218\uc5f4 \\(\\xi \\in \\mathcal{S}(P)\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert S(f,\\,P,\\,\\xi ) - J \\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(b-a < N\\delta\\)\uc778 \uc790\uc5f0\uc218 \\(N\\)\uc744 \ud0dd\ud558\uc790. \uadf8\ub7ec\uba74\n\\[\\left\\lVert P_N \\right\\rVert = \\frac{b-a}{N} < \\delta\\]\n\uc774\ubbc0\ub85c\n\\[\\left\\lvert S(f,\\,P_N ,\\, \\xi^{(N)} ) - J \\right\\rvert < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(n > N\\)\uc774\ub77c\uace0 \ud558\uba74 \\(\\lVert P_n \\rVert < \\delta\\)\uc774\uace0\n\\[S (f,\\,P_n ,\\, \\xi^{(n)} ) = \\sum_{i=1}^n f \\left(a+ \\frac{b-a}{n} i \\right) \\frac{b-a}{n}\\]\n\uc774\ubbc0\ub85c\n\\[\\left\\lvert \\sum_{i=1}^{n} f\\left( a+ \\frac{b-a}{n} i \\right) \\frac{b-a}{n} - J \\right\\rvert < \\epsilon \\tag{14}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c\ub9c8\ub2e4 (14)\uac00 \uc131\ub9bd\ud558\ubbc0\ub85c \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec (13)\uc758 \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4. \uc608\uc81c 3.<\/span> \ud480\uc774.<\/span> \\(a = 2,\\) \\(b=5 \\)\ub85c \ub450\uace0 \uc815\ub9ac 6\uc758 \uc2dd (13)\uc744 \uc774\uc6a9\ud558\uba74 \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2d0\uc9c0\ub77c\ub3c4, \ubd88\uc5f0\uc18d\uc778 \uc810\uc774 \ub9ce\uc9c0 \uc54a\uc73c\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n \uc815\ub9ac 7. (\ubd88\uc5f0\uc18d\uc810\uc758 \uac1c\uc218\uac00 \uc720\ud55c\uc778 \ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131)<\/span><\/p>\n \\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\uc73c\uba70 \uc774 \uad6c\uac04\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\uac00 \uc720\ud55c\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n \uc99d\uba85<\/p>\n \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\ub97c \ubcc0\uc218\ub85c \ud558\ub294 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uc790.<\/p>\n \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \uacbd\uc6b0 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n \uc774\uc81c \\(f\\)\uac00 \\([a,\\,b]\\)\uc758 \uc810 \uc911\uc5d0\uc11c \uc624\uc9c1 \ud55c \uc810 \\(c\\)\uc5d0\uc11c\ub9cc \ubd88\uc5f0\uc18d\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \\(a < c < b\\)\ub77c\uace0 \ud574\ub3c4 \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\ub294\ub2e4. \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc774\ubbc0\ub85c \\([a,\\,b]\\)\uc5d0\uc11c \\(\\lvert f \\rvert\\)\uc758 \uc0c1\uacc4 \\(M > 0\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc989 \uc784\uc758\uc758 \\(x\\in[a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(-M \\le f(x) \\le M\\)\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(k\\)\uac00 \uc790\uc5f0\uc218\uc774\uace0 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\uac00 \\(k\\) \uc774\ud558\uc778 \uacbd\uc6b0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\uac00 \\((k+1)\\)\uc774\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810 \ud558\ub098\ub97c \ud0dd\ud558\uc5ec \\(c\\)\ub77c\uace0 \ud558\uc790. \ubb3c\ub860 \\(a < c < b\\)\ub77c\uace0 \uac00\uc815\ud574\ub3c4 \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\ub294\ub2e4. \uc774\ubc88\uc5d0\ub3c4 (15)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(\\delta > 0\\)\uc744 \ud0dd\ud558\uc790. \\([a,\\,c-\\delta]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\ub294 \\(k\\) \uc774\ud558\uc774\ubbc0\ub85c \\(f\\)\ub294 \\([a,\\,c-\\delta]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\([a,\\,c-\\delta]\\)\uc758 \ubd84\ud560 \\(P_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec (16)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\([c+\\delta ,\\, b]\\)\uc758 \ubd84\ud560 \\(P_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec (18)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(P = P_1 \\cup P_2\\)\ub77c\uace0 \ud558\uba74 \\(P\\)\ub294 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\uba70, \uc55e\uc5d0\uc11c\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uadf8\ub7ec\ubbc0\ub85c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec, \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\uac00 \\(n\\)\uc77c \ub54c \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\uac00 \ubb34\ud55c\uc77c\uc9c0\ub77c\ub3c4 \uc801\ubd84 \uac00\ub2a5\ud560 \uc218 \uc788\ub2e4.<\/p>\n \uc608\uc81c 4.<\/span> \ud480\uc774.<\/span> \uc801\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\ub97c \ud569\uc131\ud55c \ud568\uc218\uac00 \ud56d\uc0c1 \uc801\ubd84 \uac00\ub2a5\ud55c \uac83\uc740 \uc544\ub2c8\ub2e4. \ud558\uc9c0\ub9cc \\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \\(g\\)\uac00 \uc5f0\uc18d\uc778 \uacbd\uc6b0 \\(g\\circ f\\)\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc740 \ubcf4\uc7a5\ub41c\ub2e4.<\/p>\n \uc815\ub9ac 8. (\ud569\uc131\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131)<\/span><\/p>\n \\([a,\\,b]\\)\uc640 \\([c,\\,d]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc774\uba70 \\(g\\)\uac00 \\([c,\\,d]\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \\(g\\)\uac00 \\([c,\\,d]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba70 \uc784\uc758\uc758 \\(x\\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\in [c,\\,d]\\)\uc774\uba74 \\(g\\circ f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n \uc99d\uba85<\/p>\n \\(h=g\\circ f\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(g\\)\ub294 \\([c,\\,d]\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ubbc0\ub85c \\(\\delta_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(s\\in [c,\\,d],\\) \\(t\\in [c,\\,d]\\)\uc5d0 \ub300\ud558\uc5ec \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\ud55c\uc744 \\(M,\\) \ud558\ud55c\uc744 \\(m\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 <\/p>\n \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ub9ac\ub9cc \uc801\ubd84\uc744 \uc815\uc758\ud558\uace0 \ub9ac\ub9cc \uc801\ubd84\uacfc \ub3d9\uce58\uc778 \uac1c\ub150\uc744 \uc0b4\ud3b4\ubcf4\uc558\uc73c\uba70, \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud558\ub294 \uba87 \uac00\uc9c0 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \ub610\ud55c \uad6c\ubd84\uad6c\uc801\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc801\ubd84\uac12\uc744 \uacc4\uc0b0\ud558\ub294 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc558\ub2e4. <\/p>\n \ud558\uc9c0\ub9cc \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc640 \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\uc740 \ud568\uc218\ub97c \uc27d\uac8c \uad6c\ubd84\ud558\ub294 \ubc29\ubc95\uc740 \ubc1d\ud788\uc9c0 \uc54a\uc558\uc73c\uba70, \uc801\ubd84\uac12\uc744 \uacc4\uc0b0\ud558\ub294 \uc77c\ubc18\uc801\uc778 \ubc29\ubc95 \ub610\ud55c \ubc1d\ud788\uc9c0 \uc54a\uc558\ub2e4. \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud558\ub294 \uc720\uc6a9\ud55c \ubc29\ubc95\uc73c\ub85c\ub294 \ub974\ubca0\uadf8\uc758 \uc815\ub9ac<\/span>(\ubc14\ub85c\uac00\uae30<\/a>)\uac00 \uc788\uc73c\uba70, \uc801\ubd84\uac12\uc744 \uacc4\uc0b0\ud558\ub294 \ubc29\ubc95\uc73c\ub85c\ub294 \ubbf8\uc801\ubd84\ud559\uc758 \uae30\ubcf8 \uc815\ub9ac<\/span>\uac00 \uc788\ub2e4. \uc774 \ub450 \uc815\ub9ac\ub294 \ub2e4\uc74c \ud3ec\uc2a4\ud2b8\uc5d0\uc11c \ub2e4\ub8e8\uae30\ub85c \ud558\uc790.<\/p>\n <\/p>\n <\/p>\n <\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc784\uc758\uc758 \\(x\\in[a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) > 0\\)\uc77c \ub54c, \\(x\\)\ucd95\uacfc \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504, \uadf8\ub9ac\uace0 \ub450 \uc9c1\uc120 \\(x=a,\\) \\(x=b\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \ubd80\ubd84\uc758 \ub113\uc774\ub97c \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc815\uc801\ubd84\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\uc640 \uac19\uc740 \uc815\uc758\ub294 \uc9c1\uad00\uc801\uc778 \uc815\uc758\uc774\uba70 \uc5f0\uc18d\ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub9cc \uc815\uc758\ub418\ubbc0\ub85c \ub300\ub2e8\ud788 \ud611\uc18c\ud558\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ub9ac\ub9cc \uc801\ubd84\uc744 \uc5c4\ubc00\ud558\uac8c \uc815\uc758\ud558\uace0, \uc801\ubd84 \uac00\ub2a5\uc131\uacfc \uc815\uc801\ubd84\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uad6c\ubd84\uad6c\uc801\ubc95 \\(I = [a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[292,288,294,304,308,295,91,92,290,300,310,303,307,293,296,291,289,299,309,297,301,305,298,287,302,306],"class_list":["post-1976","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-darboux-integral","tag-definite-integral","tag-integration-by-summation","tag-lower-integral","tag-lower-sum","tag-norm","tag-partition","tag-refinement","tag-riemann-integral","tag-riemann-sum","tag-riemanns-theorem","tag-upper-integral","tag-upper-sum","tag-293","tag-296","tag-291","tag-289","tag-299","tag-309","tag-297","tag-301","tag-305","tag-298","tag-287","tag-302","tag-306"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1976","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=1976"}],"version-history":[{"count":88,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1976\/revisions"}],"predecessor-version":[{"id":4862,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1976\/revisions\/4862"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1976"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1976"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1976"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\\[\\sum_{i=1}^{n} f\\left( x_i \\right) \\Delta x_i\\tag{1}\\]
\n\ub294 \\(n\\)\uc758 \uac12\uc774 \ud074\uc218\ub85d \\(A\\)\uc5d0 \uac00\uae4c\uc6cc\uc9c4\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \uadf9\ud55c
\n\\[\\lim_{n\\to\\infty}\\sum_{i=1}^{n} f\\left( x_i \\right) \\Delta x_i\\tag{2}\\]
\n\ub97c \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc815\uc801\ubd84\uc774\ub77c\uace0 \ubd80\ub974\uace0
\n\\[\\int_a^b f(x)\\,dx\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774\ub7ec\ud55c \ubc29\ubc95\uc73c\ub85c \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\ud568\uc218 \\(f\\)\uc758 \uc815\uc801\ubd84\uc744 \uc815\uc758\ud55c \uac83\uc744 \uad6c\ubd84\uad6c\uc801\ubc95<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n
\n\\[f(x) =
\n\\begin{cases}
\n1 \\quad &\\text{if} \\,\\,\\, x \\in \\mathbb{Q} \\\\[6pt]
\n0 \\quad &\\text{if} \\,\\,\\, x \\notin \\mathbb{Q}
\n\\end{cases}\\]
\n\uc774\uace0 \\([a,\\,b] = [2,\\,5]\\)\ub77c\uace0 \ud558\uba74
\n\\[\\lim_{n\\to\\infty}\\sum_{i=1}^{n} f\\left( x_i \\right) \\Delta x_i = 3\\]
\n\uc774\uc9c0\ub9cc \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc640 \\(x\\)\ucd95 \uc0ac\uc774\uc758 \ub113\uc774\ub294 \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4. [\uc124\ub839 \uce21\ub3c4\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uc5ec \ub113\uc774\uc758 \uac1c\ub150\uc744 \ud655\uc7a5\ud55c\ub2e4 \ud558\ub354\ub77c\ub3c4, \uc720\ub9ac\uc218 \uc9d1\ud569\uc758 \uae38\uc774\ub294 \\(0\\)\uc774\ubbc0\ub85c \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc640 \\(x\\)\ucd95 \uc0ac\uc774\uc758 \ub113\uc774\ub294 \\(0\\)\uc774\ub2e4.]<\/p>\n\ub9ac\ub9cc \uc801\ubd84<\/h3>\n
\n\\[a = x_0 < x_1 < x_2 < \\cdots < x_{n-1} < x_n = b\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \uc9d1\ud569\n\\[P = \\left\\{ x_0 ,\\, x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_n \\right\\}\\]\n\uc744 \\([a,\\,b]\\)\uc758 \ubd84\ud560<\/span>(partition)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ubd84\ud560 \\(P\\)\uc5d0 \uc758\ud558\uc5ec \\([a,\\,b]\\)\ub294 \\(n\\)\uac1c\uc758 \uc18c\uad6c\uac04<\/span>(subinterval)
\n\\[[x_0 ,\\, x_1 ] ,\\,\\, [x_1 ,\\,x_2 ],\\,\\, \\cdots ,\\,\\, [x_{n-1} ,\\, x_n ] \\]
\n\uc73c\ub85c \ucabc\uac1c\uc5b4 \uc9c4\ub2e4. \\(i\\)\ubc88\uc9f8 \uc18c\uad6c\uac04\uc758 \uae38\uc774\ub97c \\(\\Delta x_i\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989
\n\\[\\Delta x_i = x_i – x_{i-1} \\quad (i=1,\\,2,\\,\\cdots,\\,n)\\]
\n\uc774\ub2e4. \uac01 \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\uc5d0\uc11c \uc810 \\(\\xi_i\\)\ub97c \ud558\ub098\uc529 \ud0dd\ud558\uc5ec \uc720\ud55c\uc218\uc5f4
\n\\[\\xi = \\left\\{ \\xi _1 ,\\, \\xi_2 ,\\, \\cdots ,\\, \\xi_n \\right\\}\\]
\n\uc744 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4. \uc774 \uc218\uc5f4 \\(\\xi\\)\ub97c \ud45c\uc9d1\uc218\uc5f4<\/span>(sample)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\ub54c
\n\\[S ( f,\\,P,\\,\\xi ) = \\sum_{i=1}^{n} f(\\xi_i ) \\Delta x_i \\tag{3}\\]
\n\ub97c \u2018\ubd84\ud560 \\(P\\)\uc640 \ud45c\uc9d1\uc218\uc5f4 \\(\\xi\\)\uc5d0 \uc758\ud55c \\(f\\)\uc758 \ub9ac\ub9cc \ud569<\/span>(Riemann sum)\u2019\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. (3)\uc744
\n\\[\\sum_{ P } f\\left( \\xi_i \\right) \\Delta x_i \\]
\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n
\n\\[\\lVert P \\rVert = \\max \\left\\{ \\Delta x_i \\,\\vert\\, i = 1,\\,2,\\, \\cdots ,\\, n\\right\\}\\]
\n\uc774\ub2e4.<\/p>\n
\n\\[\\lim_{\\lVert P \\rVert \\to 0} \\sum_{P} f\\left(\\xi_i \\right) \\Delta x_i =J\\tag{7}\\]
\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \u2018\\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0 \\(J\\)\ub97c \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ub9ac\ub9cc \uc801\ubd84<\/span>(Riemann integral) \ub610\ub294 \uc815\uc801\ubd84<\/span>(definite integral) \ub610\ub294 \uc801\ubd84\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \uc0c1\ud669\uc744 \uae30\ud638\ub85c\ub294
\n\\[\\int_a^b f(x) \\,dx = J\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d (7)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(J\\)\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\uba74 \u2018\\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.\n<\/p>\n<\/div>\n
\n\uad6c\uac04 \\([2,\\,5]\\)\uc5d0\uc11c \ub2e4\uc74c \ud568\uc218\uc758 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud558\uc2dc\uc624.
\n\\[f(x) =
\n\\begin{cases}
\n1 \\quad &\\text{if} \\,\\,\\, x \\in \\mathbb{Q} \\\\[6pt]
\n0 \\quad &\\text{if} \\,\\,\\, x \\notin \\mathbb{Q}
\n\\end{cases}\\]\n<\/p>\n
\n\uc8fc\uc5b4\uc9c4 \ud568\uc218 \\(f\\)\uac00 \\([2,\\,5]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \uadf8 \uc801\ubd84\uac12\uc744 \\(J\\)\ub77c\uace0 \ud558\uc790.<\/p>\n
\n\\[x_i = 2 + \\frac{3i}{n} ,\\,\\, i=0,\\,1,\\,2,\\,\\cdots,\\,n\\]
\n\uc774\ub77c\uace0 \ud558\uba74 \\(P = \\left\\{ x_0 ,\\, x_1 ,\\, \\cdots ,\\, x_n \\right\\}\\)\uc740 \\([2,\\,5]\\)\uc758 \ubd84\ud560\uc774\uba70 \\(\\lVert P \\rVert < \\delta\\)\uc774\ub2e4. \\(P\\)\uc5d0 \uc758\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c4 \uac01 \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\uc5d0\uc11c \ubb34\ub9ac\uc218 \\(\\xi_i\\)\ub97c \ud558\ub098\uc529 \ud0dd\ud558\uc5ec \uc218\uc5f4 \\(\\xi = \\left\\{ \\xi_i \\right\\}\\)\ub97c \ub9cc\ub4e4\uc790. \uadf8\ub7ec\uba74\n\\[S(f,\\,P,\\,\\xi ) = \\sum_{i=1}^n f(\\xi _i ) \\Delta x_i = \\sum_{i=1}^n 0 \\cdot \\Delta x_i = 0\\]\n\uc774\ubbc0\ub85c\n\\[\\lvert S(f,\\,P,\\,\\xi ) - J \\rvert = J \\ge \\epsilon\\]\n\uc774\ub2e4. \uc989 \uc544\ubb34\ub9ac \uc791\uc740 \\(\\delta > 0\\)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4 \ud558\ub354\ub77c\ub3c4 \ub9ac\ub9cc \ud569\uacfc \\(J\\)\uc758 \ucc28\uc774\uac00 \\(\\epsilon\\)\ubcf4\ub2e4 \uc791\uc544\uc9c0\uc9c0 \ubabb\ud558\ubbc0\ub85c \\(J\\)\ub294 \\(f\\)\uc758 \ub9ac\ub9cc \uc801\ubd84\uac12\uc774 \uc544\ub2c8\ub2e4.<\/p>\n
\n\\[S(f,\\,P,\\,\\xi ) = \\sum_{i=1}^n f(\\xi _i ) \\Delta x_i \\ge \\sum_{i=1}^n 1 \\cdot \\frac{3}{n} = 3\\]
\n\uc774\ubbc0\ub85c
\n\\[\\lvert S(f,\\,P,\\,\\xi ) – J \\rvert = \\lvert 3-J \\rvert = 3-J \\ge \\epsilon\\]
\n\uc774\ub2e4. \uc774\ubc88\uc5d0\ub3c4 \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc544\ubb34\ub9ac \uc791\uc740 \\(\\delta > 0\\)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4 \ud558\ub354\ub77c\ub3c4 \ub9ac\ub9cc \ud569\uacfc \\(J\\)\uc758 \ucc28\uc774\uac00 \\(\\epsilon\\)\ubcf4\ub2e4 \uc791\uc544\uc9c0\uc9c0 \ubabb\ud55c\ub2e4. \uc989 \\(J\\)\ub294 \\(f\\)\uc758 \ub9ac\ub9cc \uc801\ubd84\uac12\uc774 \uc544\ub2c8\ub2e4.<\/p>\n
\n<\/span><\/p>\n<\/div>\n\ub2e4\ub974\ubd80 \uc801\ubd84<\/h3>\n
\n\\[ \\begin{align}
\nM_i (f,\\,P) &= \\sup \\left\\{ f(x) \\,\\vert\\, x\\in [ x_{i-1} ,\\, x_i ] \\right\\} ,\\\\[8pt]
\nm_i (f,\\,P) &= \\inf \\left\\{ f(x) \\,\\vert\\, x\\in [ x_{i-1} ,\\, x_i ] \\right\\}
\n\\end{align}\\]
\n\uc774\ub2e4. \ubb38\ub9e5\uc0c1 \ud568\uc218 \\(f\\)\uc640 \ubd84\ud560 \\(P\\)\uac00 \uba85\ud655\ud560 \uacbd\uc6b0 \\(M_i (f,\\,P)\\)\uc640 \\(m_i (f,\\,P)\\)\ub97c \uac01\uac01 \uac04\ub2e8\ud558\uac8c \\(M_i\\)\uc640 \\(m_i\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \uc774\ub54c
\n\\[\\begin{align}
\nU (f,\\,P) &= \\sum_{P} M_i \\Delta x_i = \\sum_{i=1}^{n} M_i \\Delta x_i ,\\\\[8pt]
\nL (f,\\,P) &= \\sum_{P} m_i \\Delta x_i = \\sum_{i=1}^{n} m_i \\Delta x_i
\n\\end{align}\\]
\n\ub97c \uac01\uac01 \\([a,\\,b]\\)\uc5d0\uc11c \\(P\\)\uc5d0 \uc758\ud55c \\(f\\)\uc758 \uc0c1\ud569<\/span>(upper sum), \\([a,\\,b]\\)\uc5d0\uc11c \\(P\\)\uc5d0 \uc758\ud55c \\(f\\)\uc758 \ud558\ud569<\/span>(lower sum)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n
\n\\[L(f,\\,P_1 ) \\le L(f,\\,P_3 ) \\le U(f,\\,P_3 ) \\le U(f,\\,P_2 )\\]
\n\uc774\ubbc0\ub85c, \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc784\uc758\uc758 \uc0c1\ud569\uc740 \uc784\uc758\uc758 \ud558\ud569\ubcf4\ub2e4 \ud06c\uac70\ub098 \ub610\ub294 \ub458\uc774 \uac19\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\ud569\ub4e4\uc758 \ubaa8\uc784\uc740 \uc544\ub798\ub85c \uc720\uacc4\uc774\uba70, \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ud558\ud569\ub4e4\uc758 \ubaa8\uc784\uc740 \uc544\ub798\ub85c \uc720\uacc4\uc774\ub2e4. \ubd84\ud560 \\(P\\)\ub97c \ubcc0\uc218\ub85c \ud588\uc744 \ub54c, \\(f\\)\uc758 \uc0c1\ud569\uc758 \ud558\ud55c\uc744 \\(f\\)\uc758 \uc0c1\uc801\ubd84\uc774\ub77c\uace0 \ubd80\ub974\uba70, \\(f\\)\uc758 \ud558\ud569\uc758 \uc0c1\ud55c\uc744 \\(f\\)\uc758 \ud558\uc801\ubd84\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84<\/span>(upper integral)\uc740
\n\\[\\overline{\\int_a^b} f(x) \\,dx = \\inf \\left\\{ U(f,\\,P) \\,\\vert\\, P \\in \\mathcal{P}([a,\\,b])\\right\\}\\]
\n\ub85c \uc815\uc758\ub418\uba70, \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ud558\uc801\ubd84<\/span>(lower integral)\uc740
\n\\[\\underline{\\int_a^b} f(x) \\,dx = \\sup \\left\\{ L(f,\\,P) \\,\\vert\\, P \\in \\mathcal{P}([a,\\,b])\\right\\}\\]
\n\ub85c \uc815\uc758\ub41c\ub2e4.<\/p>\n
\n\\[\\overline{\\int_a^b} f(x) \\,dx = \\underline{\\int_a^b} f(x)\\,dx\\]
\n\uc774\uba74 \u2018\\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5<\/span>\ud558\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\ub54c \uc0c1\uc801\ubd84\uac12\uc744 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ub2e4\ub974\ubd80 \uc801\ubd84\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub2e4\ub974\ubd80 \uc801\ubd84\uc744 \ub9ac\ub9cc \uc801\ubd84\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c
\n\\[\\int_a^b f(x)\\,dx\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud560 \ub54c
\n\\[ \\int_a^b f(x)\\,dx = \\overline{\\int_a^b} f(x) \\,dx = \\underline{\\int_a^b} f(x)\\,dx \\]
\n\uc774\ub2e4.<\/p>\n<\/div>\n
\n\\[U(f,\\,P) – L(f,\\,P) < \\epsilon\\tag{8}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n
\n\\[\\overline{\\int_a^b} f(x)\\,dx = J\\]
\n\uc774\ubbc0\ub85c \ud558\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec
\n\\[J \\le U(f,\\,P_1) < J + \\frac{\\epsilon}{2}\\tag{9}\\]\n\uc778 \ubd84\ud560 \\(P_1\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec\n\\[J - \\frac{\\epsilon}{2} < L(f,\\,P_2) \\le J\\tag{10}\\]\n\uc778 \ubd84\ud560 \\(P_2\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(P = P_1 \\cup P_2\\)\ub77c\uace0 \ud558\uba74 \\(P\\)\ub294 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\uace0, (9)\uc640 (10)\uc5d0 \uc758\ud558\uc5ec\n\\[U(f,\\,P) - L(f,\\,P) \\le U(f,\\,P_2 ) + L(f,\\,P_1 ) < \\epsilon\\]\n\uc774\ub2e4.<\/p>\n
\n\\[0 \\le \\overline{\\int_a^b} f(x)\\,dx – \\underline{\\int_a^b}f(x)\\,dx \\le U(f,\\,P) – L(f,\\,P) < \\epsilon\\]\n\uc774\uace0, \\(\\epsilon\\)\uc774 \uc784\uc758\uc758 \uc591\uc218\uc774\ubbc0\ub85c\n\\[ \\overline{\\int_a^b} f(x)\\,dx = \\underline{\\int_a^b}f(x)\\,dx \\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.\n<\/span><\/p>\n<\/div>\n
\n\\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e8\uc870\ud568\uc218\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/p>\n
\n\\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. [\uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \\(f\\)\uac00 \uac10\uc18c\ud568\uc218\ub77c\uba74 \\(-f\\)\ub294 \uc99d\uac00\ud568\uc218\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.] \ub610\ud55c \\(f(a) < f(b)\\)\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud558\uba74 \ucda9\ubd84\ud558\ub2e4. [\uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \\(f(a) \\ge f(b)\\)\ub77c\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc0c1\uc218\ud568\uc218\uac00 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.]<\/p>\n\ub9ac\ub9cc \uc801\ubd84\uacfc \ub2e4\ub974\ubd80 \uc801\ubd84\uc758 \uad00\uacc4<\/h3>\n
\n\\[\\lvert S(f,\\,P,\\,\\xi ) – J \\rvert < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4. \uc774\ub54c \\(J\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ub2e4\ub974\ubd80 \uc801\ubd84\uac12\uc774 \ub41c\ub2e4.<\/p>\n<\/div>\n
\n\\[J – \\epsilon < L(f,\\,P_\\epsilon ) \\quad \\text{and} \\quad U(f,\\,P_\\epsilon ) < J + \\epsilon\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(P\\)\uac00 \\(P_\\epsilon\\)\uc758 \uc138\ub828\ubd84\ud560\uc774\uace0 \\(\\xi\\in\\mathcal{S}(P)\\)\uc774\uba74\n\\[J - \\epsilon < L(f,\\,P_\\epsilon ) \\le (f,\\,P) \\le S(f,\\,P,\\,\\xi ) \\le U(f,\\,P) \\le U(f,\\,P_\\epsilon ) < J + \\epsilon\\]\n\uc774\ubbc0\ub85c \\(\\lvert S(f,\\,P,\\,\\xi ) - J \\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n
\n\\[\\lvert S(f,\\,P,\\,\\xi ) – J \\rvert < \\frac{\\epsilon}{3} \\tag{11}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(P = \\left\\{ x_0 ,\\, x_1 ,\\, \\cdots ,\\, x_n \\right\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc774\ubbc0\ub85c \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\uc5d0 \uc810 \\(t_i ,\\) \\(u_i \\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[f(t_i ) - f(u_i ) > M_i (f,\\,P) – m_i (f,\\,P) – \\frac{\\epsilon}{3(b-a)}\\]
\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub530\ub77c\uc11c (11)\uc5d0 \uc758\ud558\uc5ec
\n\\[\\begin{align}
\nU(f,\\,P) &- L(f,\\,P) = \\sum_{i=1}^n (M_i – m_i ) \\Delta x_i \\\\[6pt]
\n& < \\sum_{i=1}^n (f(t_i ) - f(u_i ))\\Delta x_i + \\frac{\\epsilon}{3(b-a)} \\sum_{i=1}^n \\Delta x_i \\\\[6pt]\n&\\le \\left\\lvert \\sum_{i=1}^n f(t_i ) \\Delta x_i - J \\right\\rvert + \\left\\lvert J - \\sum_{i=1}^n f(u_i ) \\Delta x_i \\right\\rvert + \\frac{\\epsilon}{3(b-a)} \\sum_{i=1}^n \\Delta x_i \\\\[6pt]\n& < \\frac{\\epsilon}{3} + \\frac{\\epsilon}{3} + \\frac{\\epsilon}{3} = \\epsilon\n\\end{align}\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.\n<\/span><\/p>\n<\/div>\n
\n\\[P_1 = \\left\\{ y_0 ,\\, y_1 ,\\, \\cdots ,\\, y_m \\right\\}\\]
\n\uc774 \uc874\uc7ac\ud558\uc5ec
\n\\[J – \\frac{\\epsilon}{4} < L(f,\\,P_1 ) \\le J \\le U(f,\\,P_1 ) < J + \\frac{\\epsilon}{4}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\([a,\\,b]\\)\uc5d0\uc11c \\(\\lvert f \\rvert\\)\uc758 \uc0c1\ud55c\uc744 \\(M\\)\uc774\ub77c\uace0 \ud558\uc790. \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(M > 0\\)\uc774\uace0 \\(m \\ge 2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
\n\\[0 < \\delta < \\min \\left\\{ \\frac{\\epsilon}{2M(m-1)} ,\\, \\Delta y_1 ,\\, \\Delta y_2 ,\\, \\cdots ,\\, \\Delta y_m \\right\\}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(\\delta\\)\ub97c \ud0dd\ud558\uc790.<\/p>\n
\n\\[m_j = \\inf f(x) ,\\,\\, M_j = \\sup f(x) ,\\,\\, x\\in [y_{j-1} ,\\, y_j ] ,\\,\\, \\xi_i \\in [x_{i-1} ,\\, x_i]\\]
\n\uc5d0 \ub300\ud558\uc5ec
\n\\[m_j \\le f(\\xi_i ) \\le M_j\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n
\n\\[y_{j-1} < x_{i-1} < y_j < x_i < y_{j+1}\\]\n\uc774\ub2e4. \uc774\ub7ec\ud55c \uc0c1\ud669\uc740 \ub9ce\uc544\uc57c \\((m-1)\\)\ubc88 \ubc1c\uc0dd\ud55c\ub2e4. \ud55c\ud3b8\n\\[f(\\xi_i ) (x_i - x_{i-1}) = f(\\xi_i )(x_i -y_j ) + f(\\xi_i )(y_j - x_{i-1})\\]\n\uc774\ubbc0\ub85c, \\(\\xi_i \\in [x_{i-1} ,\\, y_j ]\\)\uc77c \ub54c\uc5d0\ub294 \\(m_j \\le f(\\xi_i ) \\le M_j\\)\uc774\uace0, \\(\\xi_i \\in [ y_j ,\\, x_i ]\\)\uc77c \ub54c\uc5d0\ub294\n\\[f(\\xi_i ) (x_i - y_j ) < f(\\xi_i ) \\delta < \\frac{f(\\xi_i )\\epsilon}{2M(m-1)} \\le \\frac{\\epsilon}{2(m-1)}\\]\n\uc774 \uc131\ub9bd\ud558\ub294\ub370, \uc774\uac83\uc740 \ub9ce\uc544\uc57c \\((m-1)\\)\ubc88 \ubc1c\uc0dd\ud55c\ub2e4. \ub530\ub77c\uc11c \uc774\ub7ec\ud55c \ucc28\ub4e4\uc758 \ud569\uc740 \\(\\epsilon \/ 2\\)\ubcf4\ub2e4 \uc791\ub2e4. \uc774\ub85c\uc368 \\(\\xi = \\left\\{ \\xi_i \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\lvert S(f,\\,P,\\, \\xi ) - J \\rvert \\le (U(f,\\,P_1 ) - L(f,\\,P_1 )) + \\frac{\\epsilon}{2} < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/span><\/p>\n<\/div>\n\n
\n\\(a < c < b\\)\uc778 \uacbd\uc6b0, \\(\\lVert P \\rVert < \\delta\\)\uc774\uba74\uc11c \\(c\\)\uac00 \\(P\\)\uc5d0 \uc758\ud574 \ub9cc\ub4e4\uc5b4\uc9c4 \ud55c \uc18c\uad6c\uac04\uc758 \uc548\ucabd\uc5d0 \ub193\uc774\ub3c4\ub85d \ubd84\ud560\n\\[P = \\left\\{ x_0 ,\\, x_1 ,\\, \\cdots ,\\, x_n \\right\\}\\]\n\uc744 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4. \\(c = a\\)\uc774\uac70\ub098 \\(c = b\\)\uc778 \uacbd\uc6b0, \uadf8\ub0e5 \\(\\lVert P \\rVert < \\delta\\)\uc778 \ubd84\ud560 \\(P\\)\ub97c \uad6c\uc131\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(c\\)\ub294 \\(P\\)\uc758 \uc18c\uad6c\uac04 \uc911 \ub2e8 \ud558\ub098\uc5d0\ub9cc \uc18d\ud55c\ub2e4. \uadf8 \uc18c\uad6c\uac04\uc744 \\([x_{j-1} ,\\, x_j ]\\)\ub77c\uace0 \ud558\uc790.<\/p>\n
\n\\[\\sum_{i\\ne j} f(\\xi_i ) \\Delta x_i – J\\]
\n\ub294 \uc720\ud55c\uc778 \uac12\uc774\ub2e4. \\(c \\in [x_{j-1} ,\\, x_j ]\\)\uc774\uace0, \\(k\\,\\to\\,\\infty\\)\uc77c \ub54c \\(c_{n_k} \\,\\to\\,c\\)\uc774\uba70 \\(f(c_{n_k}) \\,\\to\\, \\infty\\)\uc774\uace0, \\(\\left\\{c_{n_k}\\right\\}\\)\uc758 \ud56d \uc911\uc5d0\uc11c \ubb34\uc218\ud788 \ub9ce\uc740 \ud56d\uc774 \\([x_{j-1} ,\\, x_j ]\\)\uc5d0 \uc18d\ud558\ubbc0\ub85c, \uc18c\uad6c\uac04 \\([x_{j-1} ,\\, x_j ]\\)\uc5d0 \uc18d\ud558\ub294 \uc810 \\(\\xi_j\\) \uc911\uc5d0\uc11c
\n\\[f(\\xi_j ) \\Delta x_j + \\sum_{i\\ne j} f(\\xi_i ) \\Delta x_i – J > 1\\]
\n\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4. [\uc0ac\uc2e4\uc740 \\(c_{n_k}\\) \uc911 \\(c\\)\uc5d0 \ucda9\ubd84\ud788 \uac00\uae4c\uc6b4 \ud56d\uc744 \ud0dd\ud558\uc5ec \\(\\xi_j\\)\ub85c \ub450\uba74 \ub41c\ub2e4.] \uc774\uc81c \\(i=1,\\,2,\\,\\cdots,\\,n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\xi_i\\)\uac00 \uc815\uc758\ub418\uc5c8\ub2e4. \\(\\xi = \\left\\{ \\xi_i \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[\\begin{align}
\n \\left\\lvert S(f,\\,P,\\,\\xi ) – J \\right\\rvert &= \\left\\lvert \\sum_{i=1}^n f(\\xi_j ) \\Delta x_j – J \\right\\rvert \\\\[6pt]
\n&= f(\\xi_j ) \\Delta x_j + \\sum_{i\\ne j} f(\\xi_i ) \\Delta x_i – J > \\epsilon
\n\\end{align}\\]
\n\uc774\ub2e4. \uc989 \\(\\lVert P \\rVert < \\delta\\)\uc778 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud588\uc74c\uc5d0\ub3c4 \ubd88\uad6c\ud558\uace0 \ub9ac\ub9cc \ud569\uacfc \\(J\\)\uc758 \ucc28\uc774\ub294 \\(1\\)\ubcf4\ub2e4 \uc791\uc544\uc9c0\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4.\n<\/span><\/p>\n<\/div>\n\uc801\ubd84 \uac00\ub2a5\uc131\uacfc \uad00\ub828\ub41c \uba87 \uac00\uc9c0 \uc608\uc81c<\/h3>\n
\n\\[\\lvert s – t \\rvert < \\delta \\quad \\Rightarrow \\quad \\lvert f(s) - f(t) \\rvert < \\frac{\\epsilon}{b-a} \\tag{12}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(\\lVert P \\rVert < \\delta\\)\uc778 \\([a,\\,b]\\)\uc758 \ubd84\ud560\n\\[P = \\left\\{ x_0 ,\\, x_1 ,\\, \\cdots ,\\, x_n \\right\\}\\]\n\uc744 \ud0dd\ud558\uc790. \\(P\\)\uc758 \uac01 \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\ub294 \ub2eb\ud78c \uad6c\uac04\uc774\ubbc0\ub85c \\(f\\)\ub294 \uc774 \uc18c\uad6c\uac04\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \uc989 \uc774 \uc18c\uad6c\uac04\uc758 \ub450 \uc810 \\(s_i\\)\uc640 \\(t_i\\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[\\begin{align}\nf(s_i ) &= \\max \\left\\{ f(x) \\,\\vert\\, x\\in [x_{i-1} ,\\, x_i ] \\right\\} = M_i (f,\\, P) \\\\[8pt]\nf(t_i ) &= \\min \\left\\{ f(x) \\,\\vert\\, x\\in [x_{i-1} ,\\, x_i ] \\right\\} = m_i (f,\\, P) \n\\end{align}\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7f0\ub370 \\(\\lvert s_i - t_i \\rvert < \\delta\\)\uc774\ubbc0\ub85c (12)\uc5d0 \uc758\ud558\uc5ec\n\\[\\lvert M_i (f,\\,P) - m_i (f,\\,P) \\rvert = \\lvert f(s_i ) -f(t_i ) \\rvert < \\frac{\\epsilon}{b-a}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c\n\\[\\begin{align}\nU(f,\\,P) - L(f,\\,P) &= \\sum_{i=1}^n (M_i - m_i ) \\Delta x_i \\\\[4pt]\n& < \\frac{\\epsilon}{b-a} \\sum_{i=1}^n \\Delta x_i \\\\[4pt]\n& = \\frac{\\epsilon}{b-a} (b-a) = \\epsilon\n\\end{align}\\]\n\uc774\ubbc0\ub85c \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.\n<\/span><\/p>\n<\/div>\n
\n\\[\\int_a^b f(x)\\,d x = \\lim_{n\\to\\infty} \\sum_{i=1}^n f \\left( a+ \\frac{b-a}{n} i \\right) \\frac{b-a}{n}\\tag{13}\\]\n<\/p>\n<\/div>\n
\n\\[x_i = a+ \\frac{b-a}{n}i ,\\,\\, (i = 1,\\,2,\\,\\cdots,\\,n)\\]
\n\uc774\uace0
\n\\[P_n = \\left\\{ x_0 ,\\, x_1 ,\\, \\cdots ,\\, x_n \\right\\}\\]
\n\uc774\ub2e4. \\(P_n\\)\uc5d0 \uc758\ud574 \ub9cc\ub4e4\uc5b4\uc9c4 \uac01 \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\uc758 \uc624\ub978\ucabd \ub05d\uc810
\n\\[(\\xi^{(n)})_{i} = x_i = a+\\frac{b-a}{n} i\\]
\n\ub97c \ud0dd\ud558\uc5ec \ub9cc\ub4e0 \ud45c\uc9d1\uc218\uc5f4\uc744 \\(\\xi^{(n)} = \\left\\{ (\\xi^{(n)})_{i} \\right\\}\\)\ub77c\uace0 \ud558\uc790.<\/p>\n
\n<\/span><\/p>\n<\/div>\n
\n\ub2e4\uc74c \uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624.
\n\\[\\int_{2}^{5} x^2 \\,dx \\]\n<\/p>\n
\n\ud568\uc218 \\(f(x) = x^2\\)\uc740 \\([2,\\,5]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c, \uc774 \uad6c\uac04\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc815\ub9ac 6\uc5d0 \uc758\ud558\uc5ec \uad6c\ubd84\uad6c\uc801\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc801\ubd84\uac12\uc744 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\n
\n\\[\\begin{align}
\n\\int_2^5 x^2 \\,dx
\n&= \\lim_{n\\to\\infty} \\sum_{i=1}^{n} \\left( 2+ \\frac{3}{n}i \\right)^2 \\cdot \\frac{3}{n} \\\\[4pt]
\n&= \\lim_{n\\to\\infty} \\sum_{i=1}^{n} \\left( 4+ \\frac{12}{n} i + \\frac{9}{n^2} i^2 \\right) \\cdot \\frac{3}{n} \\\\[4pt]
\n&= \\lim_{n\\to\\infty} \\sum_{i=1}^{n} \\left( \\frac{12}{n} + \\frac{36}{n^2}i + \\frac{27}{n^3} i^2 \\right) \\\\[4pt]
\n&= \\lim_{n\\to\\infty} \\left( 12 + \\frac{36}{n^2}\\cdot \\frac{n(n+1)}{2} + \\frac{27}{n^3} \\cdot \\frac{n(n+1)(2n+1)}{6} \\right)\\\\[6pt]
\n&= 12 + 18 + 9 = 39
\n\\end{align}\\]
\n\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n
\n\\[\\delta < \\min\\left\\{ \\frac{\\epsilon}{12 M} ,\\, c-a ,\\, b-c \\right\\}\\tag{15}\\]\n\uc778 \\(\\delta > 0\\)\uc744 \ud0dd\ud558\uc790. \\(f\\)\ub294 \\([a,\\, c-\\delta ]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc774 \uad6c\uac04\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\([a,\\, c-\\delta ]\\)\uc758 \ubd84\ud560 \\(P_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec
\n\\[U(f,\\,P_1 ) – L(f,\\,P_1 ) < \\frac{\\epsilon}{3}\\tag{16}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(f\\)\ub294 \\([c+\\delta,\\, c]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\uba70, \\([c+\\delta,\\, c]\\)\uc758 \ubd84\ud560 \\(P_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[U(f,\\,P_2 ) - L(f,\\,P_2 ) < \\frac{\\epsilon}{3}\\tag{17}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(P = P_1 \\cup P_2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(P\\)\ub294 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\uba70\n\\[\\begin{align}\nU(f,\\,P) - L(f,\\,P)\n&< U(f,\\,P_1 ) + 2\\delta \\cdot M + U(f,\\,P_2 ) \\\\[8pt]&\\quad - \\left[ L(f,\\,P_1 ) - 2\\delta \\cdot M - L(f,\\,P_2 ) \\right] \\\\[8pt]\n&= (U(f,\\,P_1 ) - L(f,\\,P_1 )) + 4\\delta \\cdot M \\\\[8pt]&\\quad + (U(f,\\,P_2 ) - L(f,\\,P_2 ) ) \\\\[6pt]\n&< \\frac{\\epsilon}{3} + 4\\cdot \\frac{\\epsilon}{12 M} \\cdot M + \\frac{\\epsilon}{3} = \\epsilon \\\\[6pt]\n\\end{align}\\]\n\uc774\ubbc0\ub85c \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n
\n\\[U(f,\\,P) – L(f,\\,P) < \\frac{\\epsilon}{3} + \\frac{\\epsilon}{3} + \\frac{\\epsilon}{3} = \\epsilon\\]\n\uc774\ubbc0\ub85c \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n
\n<\/span><\/p>\n<\/div>\n
\n\\(D = \\left\\{ 1\/k \\,\\vert\\, k \\in \\mathbb{N} \\right\\}\\)\uc774\ub77c\uace0 \ud558\uace0 \ud568\uc218 \\(f : [0,\\,1] \\,\\to\\, \\mathbb{R}\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
\n\\[f(x) = \\begin{cases}
\n1 \\quad & \\text{if} \\,\\, x\\in D \\\\[6pt]
\n0 \\quad & \\text{if} \\,\\, x\\notin D
\n\\end{cases}\\]
\n\uc774\ub54c \\(f\\)\uac00 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uace0, \uc801\ubd84\uac12\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n
\n\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(c = \\epsilon \/ 2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\([c,\\,1]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\ub294 \uc720\ud55c\uc774\ubbc0\ub85c \\(f\\)\ub294 \\([c,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\([c,\\,1]\\)\uc758 \ubd84\ud560\\[P_1 = \\left\\{ x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_n \\right\\}\\]\uc774 \uc874\uc7ac\ud558\uc5ec \\[U(f,\\,P_1 ) – L(f,\\,P_1 ) < \\frac{\\epsilon}{2}\\]\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(x_0 = 0\\)\uc774\ub77c\uace0 \ud558\uace0 \\[P = \\left\\{ x_0 \\right\\} \\cup P_1 = \\left\\{ x_0 ,\\, x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_n \\right\\}\\]\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74\n\\[\\begin{align}\nU(f,\\,P) - L(f,\\,P)\n&= \\sum_{i=1}^{n} (M_i (f,\\,P) - m_i (f,\\,P)) \\Delta x_i \\\\[6pt]\n&= (M_1 (f,\\,P) - m_1 (f,\\,P)) \\Delta x_1 \\\\[6pt]&\\quad + \\sum_{i=2}^{n} (M_i (f,\\,P) - m_i (f,\\,P)) \\Delta x_i \\\\[6pt]\n&= (M_1 (f,\\,P) - m_1 (f,\\,P)) (x_1 - x_0) \\\\[6pt]&\\quad + \\sum_{i=2}^{n} (M_i (f,\\,P_1 ) - m_i (f,\\,P_1 )) \\Delta x_i \\\\[6pt]\n&\\le 1\\cdot (x_1 - x_0 ) + (U(f,\\,P_1) - L(f,\\,P_1 )) \\\\[6pt]\n&< \\frac{\\epsilon}{2} + \\frac{\\epsilon}{2} = \\epsilon\n\\end{align}\\]\n\uc774\ubbc0\ub85c \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ud2b9\ud788 \ubb34\ub9ac\uc218\uc758 \uc870\ubc00\uc131\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \uc18c\uad6c\uac04\uc740 \ubb34\ub9ac\uc218\ub97c \uc6d0\uc18c\ub85c \uac00\uc9c0\ubbc0\ub85c \\(L(f,\\,P) = 0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc704 \ubd80\ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130\n\\[0 \\le L(f,\\,P) \\le U(f,\\,P) < \\epsilon\\]\n\uc744 \uc5bb\ub294\ub2e4. \uc5ec\uae30\uc11c \\(\\epsilon\\)\uc774 \uc784\uc758\uc758 \uc591\uc218\uc774\ubbc0\ub85c \uc0c1\ud569 \\(U(f,\\,P)\\)\ub294 \uc5bc\ub9c8\ub4e0\uc9c0 \\(0\\)\uc5d0 \uac00\uae4c\uc6cc\uc9c8 \uc218 \uc788\ub2e4. \uc774\ub54c \ud558\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec\n\\[\\overline{\\int_0^1} f(x) \\,dx = \\inf \\left\\{ U(f,\\,P) \\,\\vert\\, P \\in \\mathcal{P}([0,\\,1]) \\right\\} = 0\\]\n\uc774\uba70\n\\[\\underline{\\int_0^1} f(x)\\,dx \\ge L(f,\\,P) =0\\]\n\uc774\ubbc0\ub85c\n\\[\\int_0^1 f(x) \\,dx =0\\]\n\uc774\ub2e4.\n<\/span><\/p>\n<\/div>\n
\n\\[\\lvert s-t \\rvert < \\delta_1 \\quad \\Rightarrow \\quad \\lvert g(s) - g(t) \\rvert < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\delta = \\min\\left\\{ \\delta_1 ,\\, \\epsilon \\right\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \\([a,\\,b]\\)\uc758 \ubd84\ud560 \\[P = \\left\\{ x_0 ,\\, x_1 ,\\, \\cdots ,\\, x_n \\right\\}\\]\uc774 \uc874\uc7ac\ud558\uc5ec\n\\[U (f,\\,P) - L(f,\\,P ) < \\delta ^2 \\tag{18}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.\n\\[\\begin{align}\nA &= \\left\\{ i\\in \\mathbb{N} \\,\\vert\\, i \\le n ,\\, M_i (f,\\,P) - m_i (f,\\,P) < \\delta \\right\\}, \\\\[8pt]\nB &= \\left\\{ i\\in \\mathbb{N} \\,\\vert\\, i \\le n ,\\, M_i (f,\\,P) - m_i (f,\\,P) \\ge \\delta \\right\\}\n\\end{align}\\]\n\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(i\\in A\\)\uc774\uba74 \\(\\delta\\)\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\[M_i (h,\\,P) - m_i (h ,\\,P) \\le \\epsilon\\]\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n
\n\\[K = \\sup \\left\\{ \\lvert g(t) \\rvert \\,\\vert\\, m \\le t \\le M \\right\\}\\]
\n\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(i\\in B\\)\uc774\uba74 \\[M_i (h,\\,P) – m_i (h,\\,P) \\le 2K\\]\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 (18)\uc5d0 \uc758\ud558\uc5ec
\n\\[\\delta \\sum_{i\\in B} \\Delta x_i \\le \\sum_{i\\in B} ( M_i (f,\\,P) – m_i (f,\\,P)) \\Delta x_i < \\delta^2\\]\n\uc774\ubbc0\ub85c\n\\[\\sum_{i\\in B} \\Delta x_i < \\delta\\]\n\ub97c \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c\n\\[\\begin{align}\nU(h,\\,P) - L(h,\\,P)\n&= \\sum_{i\\in A} (M_i (h,\\,P) - m_i (h,\\,P)) \\Delta x_i \\\\[4pt]\n& \\quad + \\sum_{i\\in B} (M_i (h,\\,P) - m_i (h,\\,P)) \\Delta x_i \\\\[6pt]\n&< \\epsilon (b-a) + 2K \\delta \\le \\epsilon (b-a+2K)\n\\end{align}\\]\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(\\epsilon\\)\uc740 \uc784\uc758\uc758 \uc591\uc218\uc774\ubbc0\ub85c \\(\\epsilon (b-a+2K )\\) \ub610\ud55c \uc784\uc758\uc758 \uc591\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc774\ud558\uc5ec \\(h\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.\n<\/span><\/p>\n<\/div>\n\uacb0\ub860<\/h3>\n