{"id":4752,"date":"2020-06-15T22:36:03","date_gmt":"2020-06-15T13:36:03","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=4752"},"modified":"2020-06-16T09:42:59","modified_gmt":"2020-06-16T00:42:59","slug":"calculus-double-integrability-of-a-continuous-function","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-double-integrability-of-a-continuous-function\/","title":{"rendered":"\uc5f0\uc18d\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131 (\uc774\uc911\uc801\ubd84)"},"content":{"rendered":"

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc758 \uc774\uc911\uc801\ubd84\uc744 \uc815\uc758\ud558\uace0, \uc5f0\uc18d\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \uc99d\uba85\ud569\ub2c8\ub2e4. \ub9ac\ub9cc \uc801\ubd84\uc758 \uc5c4\ubc00\ud55c \uc815\uc758\uac00 \uae30\uc5b5\ub098\uc9c0 \uc54a\ub294\ub2e4\uba74 \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ub9ac\ub9cc \uc801\ubd84\uc744 \uc18c\uac1c\ud558\ub294 \uc774\uc804 \uae00(\ubc14\ub85c\uac00\uae30<\/a>)\uc744 \uba3c\uc800 \uc77d\uc5b4 \ubcf4\uae30 \ubc14\ub78d\ub2c8\ub2e4.<\/p>\n

\ub9ac\ub9cc \uc801\ubd84\uc758 \uc815\uc758<\/h2>\n

\uba3c\uc800 \uc774\uc911\uc801\ubd84\uc744 \uc815\uc758\ud558\uc790. \\(I = [a,\\,b]\\)\uc640 \\(J = [c,\\,d]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(R = I \\times J\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
\n\\[\\begin{gather}
\nP_I = \\left\\{ x_0 ,\\, x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_m \\right\\} , \\tag{1}\\\\[7pt]
\nP_J = \\left\\{ y_0 ,\\, y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_n \\right\\} \\tag{2}
\n\\end{gather}\\]
\n\uc744 \uac01\uac01 \\(I\\)\uc640 \\(J\\)\uc758 \ubd84\ud560\uc774\ub77c\uace0 \ud558\uc790. \uc5ec\uae30\uc11c (1), (2)\uc640 \uac19\uc740 \ud45c\ud604\uc5d0\uc11c\ub294 \ub2f9\uc5f0\ud788
\n\\[\\begin{gather}
\na = x_0 < x_1 < x_2 < \\cdots < x_m = b ,\\\\[7pt]\nc = y_0 < y_1 < y_2 < \\cdots < y_n = d\n\\end{gather}\\]\n\uc778 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4. \\(0 < i \\le m,\\) \\(0 < j \\le n\\)\uc778 \uc790\uc5f0\uc218 \\(i,\\) \\(j\\)\uc5d0 \ub300\ud558\uc5ec\n\\[ R_{ij} = \\left[ x_{i-1} ,\\, x_i \\right] \\times \\left[ y_{j-1} ,\\, y_j \\right] \\]\n\ub77c\uace0 \ud558\uc790. \uc774\ub54c\n\\[P = \\left\\{ R_{ij} \\,\\vert\\, i = 1,\\,2,\\, \\cdots ,\\,m \\text{ and } j = 1,\\,2,\\, \\cdots ,\\, n \\right\\}\\]\n\uc744 \\(R\\)\uc758 \ubd84\ud560<\/span>(partition)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \\(R_{ij}\\)\ub97c \\(P\\)\uc758 \uc131\ubd84\uc0ac\uac01\ud615<\/span> \ub610\ub294 \ubd80\ubd84\uc0ac\uac01\ud615<\/span>(subsquare) \uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc9c1\uad00\uc801\uc73c\ub85c \ubd84\ud560\uc774\ub780 \uc9c1\uc0ac\uac01\ud615\uc744 \uc798\ub77c\uc11c \uc5bb\uc740 \uc791\uc740 \uc9c1\uc0ac\uac01\ud615\ub4e4\uc758 \ubaa8\uc784\uc774\ub2e4.<\/p>\n

\uc774 \ud3ec\uc2a4\ud2b8\uac00 \ub05d\ub0a0 \ub54c\uae4c\uc9c0 \\(I\\)\uc640 \\(J\\)\ub294 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(R= I \\times J\\)\uc774\uba70 \\(P_I ,\\) \\(P_J ,\\) \\(P\\)\ub294 \uac01\uac01 \\(I,\\) \\(J,\\) \\(R\\)\uc758 \ubd84\ud560\uc744 \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud558\uc790.<\/p>\n

\\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uac01 \uc131\ubd84\uc0ac\uac01\ud615 \\(R_{ij}\\)\uc5d0\uc11c \ud55c \uc810\uc529 \ud0dd\ud558\uc5ec \ub9cc\ub4e0 \uc720\ud55c\uc218\uc5f4\uc744 \\(\\left\\{ c_{ij}\\right\\}\\)\ub77c\uace0 \ud558\uc790. (\\(c_{ij}\\)\uc5d0 \ucca8\uc790\uac00 \ub450 \uac1c \ubd99\uc5b4 \uc788\ub294\ub370, \uc774\ub7ec\ud55c \uc218\uc5f4\uc744 \uc774\uc911\uc218\uc5f4\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.) \uc989 \uc784\uc758\uc758 \\(i,\\) \\(j\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[c_{ij} = \\left( \\overline{x_i} ,\\, \\overline{y_j} \\right) \\in R_{ij}\\]
\n\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ubd84\ud560 \\(P\\)\uc640 \uc218\uc5f4 \\(\\left\\{ c_{ij}\\right\\}\\)\uc5d0 \ub300\ud55c \\(f\\)\uc758 \ub9ac\ub9cc \ud569<\/span>(Riemann sum)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[S \\left( f,\\,P,\\,\\left\\{ c_{ij} \\right\\} \\right)
\n=
\n\\sum_P f \\left( c_{ij} \\right) \\Delta A_{ij} = \\sum_{j=1}^n \\sum_{i=1}^m f \\left( \\overline{x_i} ,\\, \\overline{y_j} \\right) \\Delta x_i \\, \\Delta y_j .\\tag{3}\\]
\n\ubb3c\ub860 \uc5ec\uae30\uc11c
\n\\[\\begin{align}
\n\\Delta x_i &= x_i – x_{i-1} ,\\,\\,
\n\\Delta y_j = y_j – y_{j-1} ,\\\\[7pt]
\n\\Delta A_{ij} &= \\Delta x_i \\,\\Delta y_j = \\left( x_i – x_{i-1} \\right) \\left( y_j – y_{j-1} \\right)
\n\\end{align}\\]
\n\uc774\ub2e4.<\/p>\n

\ubd84\ud560 \\(P\\)\uc758 \uc131\ubd84\uc0ac\uac01\ud615\ub4e4\uc758 \ud06c\uae30\uac00 \uc791\uc744 \ub54c, \uc989 \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed \\(R\\)\ub97c \uc798\uac8c \uc790\ub97c \ub54c \ub9ac\ub9cc \ud569\uc758 \ubcc0\ud654\ub97c \uc0b4\ud3b4\ubcf4\uace0\uc790 \ud55c\ub2e4. \uc774\uac83\uc744 \uc704\ud574\uc11c\ub294 \ubd84\ud560\uc774 \uc601\uc5ed\uc744 \uc5bc\ub9c8\ub098 \uc791\uac8c \uc790\ub978 \uac83\uc778\uc9c0 \uac00\ub2a0\ud560 \uc218 \uc788\ub294 \ub3c4\uad6c\uac00 \ud544\uc694\ud558\ub2e4. \uadf8 \ub3c4\uad6c\ub85c\uc11c \ubd84\ud560\uc758 \ub178\ub984\uc744 \ub3c4\uc785\ud558\uc790. \\(\\Delta x_i\\)\ub4e4 \uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \uac12\uc744 \\(P_I\\)\uc758 \ub178\ub984<\/span>(norm)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(\\lVert P_I \\rVert\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989
\n\\[\\lVert P_I \\rVert = \\max \\left\\{ \\Delta x_i \\,\\vert\\, i = 1 ,\\,2,\\,3,\\,\\cdots,\\, m \\right\\}\\]
\n\uc774\ub2e4. \\(P_J\\)\uc5d0 \ub300\ud574\uc11c\ub3c4 \ub9c8\ucc2c\uac00\uc9c0\ub85c
\n\\[\\lVert P_J \\rVert = \\max \\left\\{ \\Delta y_j \\,\\vert\\, j = 1 ,\\,2,\\,3,\\,\\cdots,\\, n \\right\\}\\]
\n\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \\(P_I\\)\uc758 \ub178\ub984\uacfc \\(P_J\\)\uc758 \ub178\ub984 \uc911 \ub354 \ud070 \uac12\uc744 \\(P\\)\uc758 \ub178\ub984<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70 \\(\\lVert P \\rVert\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \\(P\\)\uac00 \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed\uc758 \ubd84\ud560\uc77c \ub54c, \\(P\\)\uc758 \ub178\ub984\uc774\ub780 \\(P\\)\uc758 \uc131\ubd84\uc0ac\uac01\ud615\ub4e4\uc758 \ubcc0\uc758 \uae38\uc774 \uc911 \uac00\uc7a5 \ud070 \uac83\uc744 \uc774\ub978\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\lVert P \\lVert \\,\\rightarrow\\, 0\\)\uc774\uba74 \\(P\\)\uc758 \ubaa8\ub4e0 \uc131\ubd84\uc0ac\uac01\ud615\ub4e4\uc758 \ub113\uc774\uac00 \\(0\\)\uc5d0 \uc218\ub834\ud560\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc131\ubd84\uc0ac\uac01\ud615\ub4e4\uc758 \ubcc0\uc758 \uae38\uc774 \ub610\ud55c \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc989 \\(\\lVert P \\lVert \\,\\rightarrow\\, 0\\)\uc774\uba74 \\(\\lVert P_I \\rVert \\,\\rightarrow\\, 0\\)\uc774\uace0 \\(\\lVert P_J \\rVert \\,\\rightarrow\\, 0\\)\uc774\ub2e4.<\/p>\n

\ub9cc\uc57d \\(\\lVert P \\rVert \\,\\rightarrow\\, 0\\)\uc77c \ub54c \\(c_{ij}\\)\uc758 \uc120\ud0dd\uacfc\ub294 \uc0c1\uad00 \uc5c6\uc774 \\(P\\)\uc640 \\(\\left\\{ c_{ij}\\right\\}\\)\uc5d0 \uc758\ud55c \\(f\\)\uc758 \ub9ac\ub9cc \ud569\uc774 \uc77c\uc815\ud55c \uac12 \\(V\\)\uc5d0 \uac00\uae4c\uc6cc\uc9c0\uba74 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.
\n\\[\\lim_{\\lVert P \\rVert \\,\\rightarrow\\,0} S \\left(f,\\,P,\\,\\left\\{ c_{ij} \\right\\} \\right) = V\\tag{4}\\]
\n\ub610\ub294
\n\\[\\lim_{\\lVert P \\rVert \\,\\rightarrow\\,0} \\sum_P f\\left(c_{ij}\\right) \\Delta A_{ij} = V.\\tag{5}\\]
\n\uc989 \\(f\\)\uc758 \ub9ac\ub9cc \ud569\uc774 \\(V\\)\uc5d0 \uc218\ub834\ud55c\ub2e4<\/span>\ub294 \uac83\uc740
\n\\[\\forall \\epsilon > 0 \\, \\exists \\delta > 0 \\, \\forall P \\, \\forall \\left\\{ c_{ij} \\right\\} \\,: \\,\\,
\n\\left[\\, \\lVert P \\rVert < \\delta \\,\\, \\rightarrow \\,\\, \\left\\lvert S \\left (f,\\,P,\\,\\left\\{ c_{ij} \\right\\} \\right) - V \\right\\rvert < \\epsilon \\,\\right]\\tag{6}\\]\n\ub610\ub294\n\\[\\forall \\epsilon > 0 \\, \\exists \\delta > 0 \\, \\forall P \\,: \\,\\,
\n\\left[\\, \\lVert P \\rVert < \\delta \\,\\, \\rightarrow \\,\\, \\left( \\forall \\left\\{ c_{ij} \\right\\} \\,:\\, \\left\\lvert S \\left (f,\\,P,\\,\\left\\{ c_{ij} \\right\\} \\right) - V \\right\\rvert < \\epsilon \\right) \\,\\right]\\tag{7}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4\ub294 \ub73b\uc774\ub2e4. (\ub2f9\uc5f0\ud788 (6)\uacfc (7)\uc740 \ub17c\ub9ac\uc801\uc73c\ub85c \ub3d9\uce58\uc778 \ubb38\uc7a5\uc774\ub2e4.)<\/p>\n

\uc774\uc81c \ube44\ub85c\uc18c \ub9ac\ub9cc \uc801\ubd84\uc744 \uc815\uc758\ud560 \uc900\ube44\uac00 \ub418\uc5c8\ub2e4. \\(f\\)\uac00 \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed \\(R\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc720\uacc4\uc778 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc2e4\uc218 \\(V\\)\uac00 \uc874\uc7ac\ud558\uc5ec (4)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \u2018\\(f\\)\ub294 \\(R\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\ub54c \\(V\\)\ub97c \u2018\\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uac12<\/span>\u2019\uc774\ub77c\uace0 \ubd80\ub974\uba70
\n\\[\\iint _R \\,f \\, dA = V\\tag{8}\\]
\n\ub610\ub294
\n\\[\\iint _R \\, f(x,\\,y)\\,dx\\,dy = V\\tag{9}\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\ub2e4\ub974\ubd80 \uc801\ubd84<\/h2>\n

\ub2e4\ub974\ubd80(Darboux<\/a>)\ub294 \ub9ac\ub9cc\uacfc\ub294 \uc870\uae08 \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \uc801\ubd84\uc744 \uc815\uc758\ud558\uc600\ub2e4. <\/p>\n

\\(P_I\\)\uc640 \\(P_I^*\\)\uac00 \ubaa8\ub450 \\(I\\)\uc758 \ubd84\ud560\uc774\uace0 \\(P_I \\subseteq P_I^*\\)\uc77c \ub54c \\(P_I^*\\)\ub97c \\(P_I\\)\uc758 \uc138\ub828\ubd84\ud560<\/span>(refinement)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \uc138\ub828\ubd84\ud560\uc774\ub780 \uad6c\uac04\uc744 \ub354 \uc798\uac8c \uc790\ub978 \ubd84\ud560\uc744 \ub73b\ud55c\ub2e4.<\/p>\n

\\(P_I^*\\)\uac00 \\(P_I\\)\uc758 \uc138\ub828\ubd84\ud560\uc774\uace0 \\(P_J^*\\)\uac00 \\(P_J\\)\uc758 \uc138\ub828\ubd84\ud560\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(P\\)\uac00 \\(P_I\\)\uacfc \\(P_J\\)\ub97c \uc774\uc6a9\ud558\uc5ec \ub9cc\ub4e0 \\(R\\)\uc758 \ubd84\ud560\uc774\uba70, \\(P^*\\)\uac00 \\(P_I^*\\)\uc640 \\(P_J^*\\)\ub97c \uc774\uc6a9\ud558\uc5ec \ub9cc\ub4e0 \\(R\\)\uc758 \ubd84\ud560\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(P^*\\)\ub97c \\(P\\)\uc758 \uc138\ub828\ubd84\ud560<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. (\uad6c\uac04\uc758 \ubd84\ud560\uc740 \uad6c\uac04\uc758 \uc810\ub4e4\uc758 \ubaa8\uc784\uc774\uc9c0\ub9cc, \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed\uc758 \ubd84\ud560\uc740 \uc810\ub4e4\uc758 \ubaa8\uc784\uc774 \uc544\ub2c8\ub77c \uc791\uc740 \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed\ub4e4\uc758 \ubaa8\uc784\uc774\ub2e4.)<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \uc0c1\ud569\uacfc \ud558\ud569\uc744 \uc815\uc758\ud558\uc790. \\(P\\)\uc758 \uac01 \uc131\ubd84\uc0ac\uac01\ud615 \\(R_{ij}\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\ud55c\uacfc \ud558\ud55c\uc744 \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.
\n\\[\\begin{align}
\nM_{ij}(f) &= \\sup\\left\\{ f(c_{ij}) \\,\\vert\\, c_{ij} \\in R_{ij} \\right\\} ,\\tag{10}\\\\[7pt]
\nm_{ij}(f) &= \\inf\\left\\{ f(c_{ij}) \\,\\vert\\, c_{ij} \\in R_{ij} \\right\\} .\\tag{11}
\n\\end{align}\\]<\/p>\n

\\(P\\)\uc5d0 \uc758\ud55c \\(f\\)\uc758 \uc0c1\ud569<\/span> \\(U(f,\\,P)\\)\uc640 \ud558\ud569<\/span> \\(L(f,\\,P)\\)\uc744 \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[\\begin{align}
\nU(f,\\,P) &= \\sum_P M_{ij}(f) \\Delta A_{ij} = \\sum_{j=1}^{n} \\sum_{i=1}^{m} M_{ij}(f) \\Delta A_{ij} ,\\tag{12}\\\\[7pt]
\nL(f,\\,P) &= \\sum_P m_{ij}(f) \\Delta A_{ij} = \\sum_{j=1}^{n} \\sum_{i=1}^{m} m_{ij}(f) \\Delta A_{ij} .\\tag{13}
\n\\end{align}\\]<\/p>\n

\ubd84\ud560\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec, \uc784\uc758\uc758 \ubd84\ud560 \\(P\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[L(f,\\,P) \\le U(f,\\,P)\\tag{14}\\]
\n(\uc9c1\uad00\uc801\uc73c\ub85c \uc0dd\uac01\ud574 \ubcf4\uba74, \\(f\\)\uac00 \ud56d\uc0c1 \uc591\uc22b\uac12\uc744 \uac00\uc9c8 \ub54c, \\(L(f,\\,P)\\)\uc758 \uac12\uc774 \\(f\\)\uc758 \uadf8\ub798\ud504 \uc544\ub798\ucabd\uc758 \ubd80\ud53c\ubcf4\ub2e4 \ucee4\uc9c8 \uc218 \uc5c6\uace0 \\(U(f,\\,P)\\)\ub294 \\(f\\)\uc758 \uadf8\ub798\ud504 \uc544\ub798\ucabd\uc758 \ubd80\ud53c\ubcf4\ub2e4 \uc791\uc544\uc9c8 \uc218 \uc5c6\uae30 \ub54c\ubb38\uc5d0 \uc704 \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.)
\n\ub610\ud55c \\(P^*\\)\uac00 \\(P\\)\uc758 \uc138\ub828\ubd84\ud560\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[L(f,\\,P) \\le L(f,\\,P^* ) \\le U(f,\\,P^* ) \\le U(f,\\,P)\\tag{15}\\]
\n(\uc9c1\uad00\uc801\uc73c\ub85c \uc0dd\uac01\ud574 \ubcf4\uba74, \uc704 \ubd80\ub4f1\uc2dd\uc740 \uc815\uc758\uc5ed\uc758 \uc815\uc0ac\uac01\ud615 \uc601\uc5ed\uc744 \ub354 \uc798\uac8c \uc790\ub97c \uc218\ub85d \ud558\ud569\uacfc \uc0c1\ud569\uc774 \uadf8\ub798\ud504 \uc544\ub798\ucabd\uc758 \ubd80\ud53c\uc5d0 \uac00\uae4c\uc6cc\uc9c4\ub2e4\ub294 \ub73b\uc774\ub2e4.)
\n\ub354\uc6b1\uc774 \\(P_1\\)\uacfc \\(P_2\\)\uac00 \\(R\\)\uc758 \ubd84\ud560\uc774\uace0, \\(P\\)\uac00 \\(P_1\\)\uc758 \uc138\ub828\ubd84\ud560\uc774\uba74\uc11c \ub3d9\uc2dc\uc5d0 \\(P_2\\)\uc758 \uc138\ub828\ubd84\ud560\uc77c \ub54c, \uc989 \\(P\\)\uac00 \\(P_1\\)\uacfc \\(P_2\\)\uc758 \uacf5\ud1b5\uc138\ub828\ubd84\ud560<\/span>\uc77c \ub54c(\\(P_1\\)\uacfc \\(P_2\\)\uac00 \uc5b4\ub290\uac83\uc774 \uc8fc\uc5b4\uc9c0\ub4e0 \uadf8\ub7ec\ud55c \uacf5\ud1b5\uc138\ub828\ubd84\ud560 \\(P\\)\ub294 \ud56d\uc0c1 \uc874\uc7ac\ud55c\ub2e4), \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[L(f,\\,P_1 ) \\le L(f,\\,P) \\le U(f,\\,P) \\le U(f,\\,P_2)\\tag{16}\\]
\n\uc774 \ubd80\ub4f1\uc2dd\uc740 \u201c\ud558\ud569\uc774 \uc544\ubb34\ub9ac \ucee4\uc9c4\ub2e4 \ud55c\ub4e4 \ub2e4\ub978 \uc0c1\ud569\ubcf4\ub2e4 \ucee4\uc9c8 \uc218 \uc5c6\ub2e4\u201d\ub77c\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \ub610\ud55c \uc774 \uc0ac\uc2e4\ub85c\ubd80\ud130 \uc5bb\uc744 \uc218 \uc788\ub294 \uac83\uc740, \\(f\\)\uc758 \ud558\ud569\ub4e4\uc758 \uc9d1\ud569\uc774 \uc704\ub85c \uc720\uacc4\uc774\uace0, \\(f\\)\uc758 \uc0c1\ud569\ub4e4\uc758 \uc9d1\ud569\uc774 \uc544\ub798\ub85c \uc720\uacc4\ub77c\ub294 \uc131\uc9c8\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uc758 \ud558\ud569\ub4e4\uc758 \uc9d1\ud569\uc758 \uc0c1\ud55c\uacfc \\(f\\)\uc758 \uc0c1\ud569\ub4e4\uc758 \uc9d1\ud569\uc758 \ud558\ud55c\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774\ub4e4\uc744 \uac01\uac01 \\(f\\)\uc758 \ud558\uc801\ubd84<\/span>(lower integral), \\(f\\)\uc758 \uc0c1\uc801\ubd84<\/span>(upper integral)\uc774\ub77c\uace0 \ubd80\ub974\uba70, \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. (\uccab \ubc88\uc9f8 \uc801\ubd84 \uae30\ud638\uc5d0\ub294 \ubc11\uc904\uc774 \uadf8\uc5b4\uc838 \uc788\uace0, \ub450 \ubc88\uc9f8 \uc801\ubd84 \uae30\ud638\uc5d0\ub294 \uc717\uc904\uc774 \uadf8\uc5b4\uc838 \uc788\ub2e4.)
\n\\[\\begin{align}
\nL_f &= \\sup \\left\\{ L(f,\\,P) \\,\\vert\\, P \\text{ is a partition of }R \\right\\} ,\\tag{17}\\\\[6pt]
\nU_f &= \\inf \\left\\{ U(f,\\,P) \\,\\vert\\, P \\text{ is a partition of }R \\right\\} .\\tag{18}
\n\\end{align}\\]<\/p>\n

\ub9cc\uc57d \\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc774 \uac19\uc73c\uba74, \u2018\\(f\\)\ub294 \\(R\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \\(f\\)\uc758 \uc0c1\uc801\ubd84\uac12 \\(U_f\\)\ub97c \\(f\\)\uc758 \uc801\ubd84\uac12\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.\n<\/p>\n

\ub9ac\ub9cc \ud310\uc815\ubc95<\/h2>\n

\ub2e4\ub974\ubd80 \uc801\ubd84\uc758 \uc815\uc758\ub9cc\uc73c\ub85c\ub294 \uc5b4\ub5a0\ud55c \ud568\uc218\uac00 \uc801\ubd84 \uac00\ub2a5\ud55c\uc9c0 \ud310\ubcc4\ud558\uae30 \uc27d\uc9c0 \uc54a\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \uc27d\uac8c \ud310\uc815\ud560 \uc218 \uc788\ub294 \ubc29\ubc95\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n

\uc784\uc758\uc758 \uc591\uc218 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec \\(R\\)\uc758 \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(U(f,\\,P) – L(f,\\,P) < \\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \uc989\n\\[\\forall \\epsilon > 0 \\, \\exists P \\,:\\, ( U(f,\\,P) – L(f,\\,P) < \\epsilon )\\tag{20}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uc774\ub7ec\ud55c \uac00\uc815 \ud558\uc5d0 \\(f\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \uc99d\uba85\ud574 \ubcf4\uc790.<\/p>\n

\uc591\uc218 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 (20)\uc758 \ubd80\ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84\uc744 \\(V_U\\)\ub77c\uace0 \ud558\uace0, \\(f\\)\uc758 \ud558\uc801\ubd84\uc744 \\(V_L\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc0c1\ud55c\uacfc \ud558\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec
\n\\[L(f,\\,P) \\le V_L \\le V_U \\le U(f,\\,P)\\tag{21}\\]
\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(L(f,\\,P)\\)\uc640 \\(U(f,\\,P)\\)\uc758 \ucc28\uc774\ub294 \\(\\epsilon\\) \ubbf8\ub9cc\uc774\ubbc0\ub85c (21)\uc5d0 \uc758\ud558\uc5ec
\n\\[0\\le V_U – V_L < \\epsilon\\tag{22}\\]\n\uc744 \uc5bb\ub294\ub2e4. \uc774\uac83\uc740 \\(V_U\\)\uc640 \\(V_L\\)\uc758 \ucc28\uc774\uac00 \\(\\epsilon\\)\ubcf4\ub2e4 \uc791\ub2e4\ub294 \ub73b\uc778\ub370, \\(\\epsilon\\)\uc740 \u2018\uc784\uc758\uc758\u2019 \uc591\uc218\uc774\ubbc0\ub85c, \\(V_U\\)\uc640 \\(V_L\\)\uc758 \ucc28\uc774\ub294 \\(0\\)\uc774\ub2e4. \uc989 \\(V_U = V_L\\)\uc774\ub2e4. \\(f\\)\uc758 \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc774 \uac19\uc73c\ubbc0\ub85c \\(f\\)\ub294 \\(R\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n

\uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uadf8 \uc801\ubd84\uac12\uc744 \\(V\\)\ub77c\uace0 \ud558\uc790. \uc774\ub7ec\ud55c \uac00\uc815 \ud558\uc5d0 (20)\uc774 \uc131\ub9bd\ud569\uc744 \ubcf4\uc774\uc790.<\/p>\n

\uc591\uc218 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \uc801\ubd84\uac12\uc774 \\(V\\)\uc774\ubbc0\ub85c \\(V\\)\ub294 \\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \ud558\ud569\ub4e4\uc758 \uc0c1\ud55c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(V\\)\uc5d0 \uc5bc\ub9c8\ub4e0\uc9c0 \uac00\uae4c\uc6b4 \\(f\\)\uc758 \ud558\ud569\uc744 \ucc3e\uc744 \uc218 \uc788\ub2e4. \uc989
\n\\[\\lvert V – L(f,\\,P_1 ) \\rvert < \\frac{\\epsilon}{2}\\]\n\uc778 \ubd84\ud560 \\(P_1\\)\uc744 \ucc3e\uc744 \uc218 \uc788\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c\n\\[\\lvert U(f,\\,P_2 ) - V \\rvert < \\frac{\\epsilon}{2}\\]\n\uc778 \ubd84\ud560 \\(P_2\\)\ub97c \ucc3e\uc744 \uc218 \uc788\ub2e4. \\(P_1\\)\uacfc \\(P_2\\)\uc758 \uacf5\ud1b5\uc138\ub828\ubd84\ud560 \ud558\ub098\ub97c \uc7a1\uc544\uc11c \\(P\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74\n\\[ V-\\frac{\\epsilon}{2} < L(f,\\,P_1) \\le L(f,\\,P) \\le U(f,\\,P) \\le U(f,\\,P_2 ) < V + \\frac{\\epsilon}{2}\\]\n\uc774\ubbc0\ub85c \\(U(f,\\,P) - L(f,\\,P) < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

\uc774\ub85c\uc368 \u201c\\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 (20)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4\u201d\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4. \uc774 \uc815\ub9ac\ub97c \ub9ac\ub9cc \ud310\uc815\ubc95<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. (\ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\uc815\ud558\ub294 \ubc29\ubc95\uc5d0 \ub9ac\ub9cc\uc758 \uc774\ub984\uc774 \ubd99\uc740 \uac83\uc740, \uc0ac\uc2e4\uc740 \ub2e4\ub974\ubd80 \uc801\ubd84\uacfc \ub9ac\ub9cc \uc801\ubd84\uc774 \ub3d9\uce58\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.)<\/p>\n

\uc5f0\uc18d\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131<\/h2>\n

\uc5f0\uc18d\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ub17c\ud558\uae30 \uc704\ud574\uc11c\ub294 \u2018\uade0\ub4f1\uc5f0\uc18d\u2019\uc744 \uc815\uc758\ud574\uc57c \ud55c\ub2e4. \uade0\ub4f1\uc5f0\uc18d\uc758 \uc815\uc758\uc640 \uc131\uc9c8\uc740 \uc9c0\ub09c \ud3ec\uc2a4\ud2b8 \u2018\uc5f4\ub9b0 \uc9d1\ud569\uacfc \ub2eb\ud78c \uc9d1\ud569<\/a>\u2019\uc5d0\uc11c \uc774\ubbf8 \uc0b4\ud3b4\ubcf4\uc558\uc73c\ubbc0\ub85c, \uc5ec\uae30\uc11c\ub294 \uac04\ub2e8\ud558\uac8c \uc18c\uac1c\ub9cc \ud558\uaca0\ub2e4.<\/p>\n

\\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \uc591\uc218 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec \uc591\uc218 \\(\\delta\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\lvert \\textbf{c}_1 – \\textbf{c}_2 \\rvert < \\delta\\)\uc778 \ubaa8\ub4e0 \\(\\textbf{c}_1 ,\\, \\textbf{c}_2 \\in R\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert f( \\textbf{c}_1 ) - f( \\textbf{c}_2 ) \\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(R\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

\\(R\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc778 \ud568\uc218\ub294 \ub2f9\uc5f0\ud788 \\(R\\)\uc5d0\uc11c \uc5f0\uc18d\ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(R\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218\uac00 \ud56d\uc0c1 \\(R\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc778 \uac83\uc740 \uc544\ub2c8\ub2e4. \uadf8\ub7f0\ub370 \\(R\\)\uac00 \uc720\uacc4\uc774\uace0 \ub2eb\ud78c \uc9d1\ud569\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(R\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218\ub294 \ud56d\uc0c1 \\(R\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \uc774\ub7ec\ud55c \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uc5ec \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed \\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \uc99d\uba85\ud560 \uac83\uc774\ub2e4.<\/p>\n

\\(R\\)\uac00 \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed \\(R = [a,\\,b] \\times [c,\\,d]\\)\uc774\uace0, \\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(R\\)\uc758 \ub113\uc774\ub97c \\(A\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

\ub2e4\uc74c\uc73c\ub85c \uc591\uc218 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(R\\)\uac00 \uc720\uacc4\uc774\uace0 \ub2eb\ud78c \uc9d1\ud569\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(R\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc591\uc218 \\(\\delta_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(\\lvert \\textbf{c}_1 – \\textbf{c}_2 \\rvert < \\delta_1\\)\uc778 \ubaa8\ub4e0 \\(\\textbf{c}_1 ,\\, \\textbf{c}_2 \\in R\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\lvert f( \\textbf{c}_1 ) - f( \\textbf{c}_2 ) \\rvert < \\frac{\\epsilon}{A}\\tag{23}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\delta = \\delta_1 \/ 2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\lVert P \\rVert < \\delta\\)\uc778 \\(R\\)\uc758 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud558\uc790. \\(P\\)\uc758 \uac01 \uc131\ubd84\uc0ac\uac01\ud615\uc758 \ubcc0\uc758 \uae38\uc774\ub294 \\(\\delta_1 \/ 2\\) \uc774\ud558\uc774\ubbc0\ub85c, \\(P\\)\uc758 \uac01 \uc131\ubd84\uc0ac\uac01\ud615 \\(R_{ij}\\)\uc758 \ub300\uac01\uc120\uc758 \uae38\uc774\ub294 \\(\\delta\\) \uc774\ud558\uc774\uba70, \\(R_{ij}\\) \uc548\uc5d0\uc11c \n\\[M_{ij} (f) - m_{ij}(f) < \\frac{\\epsilon}{A}\\tag{24}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub7ec\ud55c \ubd84\ud560 \\(P\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\)\uc758 \uc0c1\ud569\uacfc \ud558\ud569\uc758 \ucc28\ub97c \uad6c\ud558\uba74\n\\[\\begin{align}\nU(f,\\,P) - L(f,\\,P) &= \\sum_P \\left( M_{ij}(f) - m_{ij}(f) \\right) \\Delta A_{ij} \\\\[7pt]\n&< \\sum_P \\frac{\\epsilon}{A} \\Delta A_{ij} = \\frac{\\epsilon}{A} \\sum_P \\Delta A_{ij} \\\\[7pt]\n&= \\frac{\\epsilon}{A} \\cdot A = \\epsilon\n\\end{align}\\]\n\uc774\ubbc0\ub85c \\[U(f,\\,P) - L(f,\\,P) < \\epsilon\\tag{25}\\]\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989 (20)\uc774 \uc131\ub9bd\ud558\ubbc0\ub85c \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\(R\\)\uc5d0\uc11c \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n

\\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \ub2e4\ub974\ubd80 \uc801\ubd84\uac12\uc744 \\(V\\)\ub77c\uace0 \ud558\uc790. \uc9c0\uae08\ubd80\ud130 \\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \ub9ac\ub9cc \ud569\uc774 \\(V\\)\uc5d0 \uc218\ub834\ud568\uc744 \ubcf4\uc784\uc73c\ub85c\uc368 \\(R\\)\uc5d0\uc11c \\(f\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \uc99d\uba85\ud558\uc790.<\/p>\n

\ub2e4\uc2dc \uc591\uc218 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ubc14\ub85c \uc55e\uc5d0\uc11c\uc640 \uac19\uc740 \uc774\uc720\uc5d0 \uc758\ud558\uc5ec (23)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc591\uc218 \\(\\delta_1\\)\uc774 \uc874\uc7ac\ud558\uba70, \\(\\delta = \\delta_1 \/ 2\\)\uc77c \ub54c \\(\\lVert P \\rVert < \\delta\\)\uc778 \ubd84\ud560 \\(P\\)\uc5d0 \ub300\ud558\uc5ec (25)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 \ub2e4\ub974\ubd80 \uc801\ubd84\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec\n\\[L(f,\\,P) \\le V \\le U(f,\\,P) \\tag{26}\\]\n\uc774\ub2e4. \ud55c\ud3b8 \\(P\\)\uc758 \uac01 \uc131\ubd84\uc0ac\uac01\ud615 \\(R_{ij}\\)\uc5d0\uc11c \ud55c \uc810\uc529 \ud0dd\ud558\uc5ec \\(c_{ij} \\in R_{ij}\\)\ub77c\uace0 \ud558\uba74\n\\[L(f,\\,P) \\le S(f,\\,P,\\, \\left\\{ c_{ij} \\right\\}) \\le U(f,\\,P) \\tag{27}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. (25), (26), (27)\uc744 \uacb0\ud569\ud558\uba74\n\\[-\\epsilon < S(f,\\,P,\\,\\left\\{ c_{ij} \\right\\} ) -V < \\epsilon\\tag{28}\\]\n\uc989\n\\[\\lvert S(f,\\,P,\\,\\left\\{ c_{ij} \\right\\} ) -V \\rvert < \\epsilon\\tag{29}\\]\n\uc744 \uc5bb\ub294\ub2e4. \uc694\ucee8\ub300 \uc784\uc758\uc758 \uc591\uc218 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec \uc591\uc218 \\(\\delta\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\lVert P \\rVert < \\delta\\)\uc77c \ub54c\ub9c8\ub2e4 \\(\\left\\{ c_{ij} \\right\\}\\)\uc758 \uc120\ud0dd\uc5d0 \uc0c1\uad00 \uc5c6\uc774 (29)\uac00 \uc131\ub9bd\ud558\ubbc0\ub85c, \\(\\lVert P \\rVert \\,\\rightarrow\\,0\\)\uc77c \ub54c \\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \ub9ac\ub9cc \ud569\uc774 \\(V\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\(R\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uba70, \ub9ac\ub9cc \uc801\ubd84\uac12\uc740 \ub2e4\ub974\ubd80 \uc801\ubd84\uac12 \\(V\\)\uc640 \ub3d9\uc77c\ud558\ub2e4.\n<\/p>\n

\ucc38\uace0<\/h2>\n

\uc0ac\uc2e4\uc740 \uc5f0\uc18d\ud568\uc218\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc720\uacc4\uc778 \uc784\uc758\uc758 \ud568\uc218 \\(f\\)\uc5d0 \ub300\ud558\uc5ec, \\(f\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f\\)\uac00 \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\ud55c \uac83\uc774\ub2e4. \uc989 \ub2e4\ub974\ubd80 \uc801\ubd84 \uac00\ub2a5\uc131\uc5d0 \ub300\ud55c \ub9ac\ub9cc \ud310\uc815\ubc95\uc740 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\uc815\ud558\ub294 \ub370\uc5d0\ub3c4 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \ub9ac\ub9cc \uc801\ubd84\uc774 \ub2e4\ub974\ubd80 \uc801\ubd84\uacfc \ub3d9\uce58\ub77c\ub294 \uc0ac\uc2e4\uc744 \uc99d\uba85\ud558\ub294 \uac83\uc740 \uaf64 \ubcf5\uc7a1\ud558\ub2e4. \uc774\uc640 \uad00\ub828\ub41c \ub0b4\uc6a9\uc740 \uc774\uc804 \ud3ec\uc2a4\ud2b8 \u2018\uc815\uc801\ubd84\uc758 \uc815\uc758\u2019(\ubc14\ub85c\uac00\uae30<\/a>)\ub97c \uc77d\uc5b4\ubcf4\uae30 \ubc14\ub780\ub2e4.<\/p>\n","protected":false},"excerpt":{"rendered":"

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc758 \uc774\uc911\uc801\ubd84\uc744 \uc815\uc758\ud558\uace0, \uc5f0\uc18d\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \uc99d\uba85\ud569\ub2c8\ub2e4. \ub9ac\ub9cc \uc801\ubd84\uc758 \uc5c4\ubc00\ud55c \uc815\uc758\uac00 \uae30\uc5b5\ub098\uc9c0 \uc54a\ub294\ub2e4\uba74 \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ub9ac\ub9cc \uc801\ubd84\uc744 \uc18c\uac1c\ud558\ub294 \uc774\uc804 \uae00(\ubc14\ub85c\uac00\uae30)\uc744 \uba3c\uc800 \uc77d\uc5b4 \ubcf4\uae30 \ubc14\ub78d\ub2c8\ub2e4. \ub9ac\ub9cc \uc801\ubd84\uc758 \uc815\uc758 \uba3c\uc800 \uc774\uc911\uc801\ubd84\uc744 \uc815\uc758\ud558\uc790. \\(I = [a,\\,b]\\)\uc640 \\(J = [c,\\,d]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(R = I \\times J\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\begin{gather} P_I = \\left\\{ x_0 ,\\, x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_m…<\/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":[47,50],"tags":[384,394,292,395,290,300,184,291,289,393],"class_list":["post-4752","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","category-mathematical-analysis","tag-calculus","tag-continuous-function","tag-darboux-integral","tag-double-integral","tag-riemann-integral","tag-riemann-sum","tag-uniform-continuity","tag-291","tag-289","tag-393"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4752","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=4752"}],"version-history":[{"count":64,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4752\/revisions"}],"predecessor-version":[{"id":4819,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4752\/revisions\/4819"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=4752"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=4752"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=4752"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}