{"id":9488,"date":"2025-10-20T18:56:00","date_gmt":"2025-10-20T09:56:00","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9488"},"modified":"2026-09-27T14:52:11","modified_gmt":"2026-09-27T05:52:11","slug":"ch06-the-riemann-integral","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch06-the-riemann-integral\/","title":{"rendered":"\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84"},"content":{"rendered":"
<\/p>\n

<\/p>\n

\uc774 \uc7a5\uc5d0\uc11c\ub294 \ub9ac\ub9cc \uc801\ubd84\uc758 \uc5c4\ubc00\ud55c \uc815\uc758\uc640 \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc801\ubd84 \uac00\ub2a5\uc131\uc758 \uc870\uac74, \ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac, \uadf8\ub9ac\uace0 \uc774\uc0c1\uc801\ubd84\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\ub9ac\ub9cc \uc801\ubd84\uc758 \uc815\uc758<\/h3>\n

\uc774 \uc808\uc5d0\uc11c\ub294 \\(a<b\\)\uc778 \uc2e4\uc218 \\(a\\), \\(b\\)\ub97c \uace0\uc815\ud55c\ub2e4. \uad6c\uac04 \\([a,\\,b]\\)\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569 \\(P=\\{x_0,x_1,\\ldots,x_n\\}\\)\uc774
\n\\[a=x_0<x_1<x_2<\\cdots<x_n=b\\]
\n\ub97c \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(P\\)\ub97c \\([a,b]\\)\uc758 \ubd84\ud560<\/span>(partition)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \uac01 \uad6c\uac04
\n\\[[x_0,x_1],\\,[x_1,x_2],\\,\\ldots,\\,[x_{n-1},x_n]\\]
\n\uc744 \\(P\\)\uc5d0 \uc758\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c4 \uc18c\uad6c\uac04<\/span> \ub610\ub294 \uc131\ubd84\uad6c\uac04<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc18c\uad6c\uac04\uc758 \uae38\uc774 \uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \uac12\uc744 \\(P\\)\uc758 \ubd84\ud560\uc758 \ub178\ub984<\/span>(norm of a partition)\uc774\ub77c\uace0 \ubd80\ub974\uba70, \\(\\lVert P\\rVert\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989
\n\\[\\lVert P\\rVert=\\max_{1\\le i\\le n}(x_i-x_{i-1}).\\]
\n\ub9cc\uc57d \\(P\\)\uc640 \\(Q\\)\uac00 \\([a,b]\\)\uc758 \ubd84\ud560\uc774\uace0 \\(P\\subseteq Q\\)\uc774\uba74, \\(Q\\)\ub97c \\(P\\)\uc758 \uc138\ub828\ubd84\ud560<\/span>(refinement)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \\(P\\), \\(Q\\), \\(R\\)\uc774 \\([a,b]\\)\uc758 \ubd84\ud560\uc774\uace0 \\(P\\cup Q\\subseteq R\\)\uc774\uba74 \\(R\\)\uc744 \\(P\\)\uc640 \\(Q\\)\uc758 \uacf5\ud1b5\uc138\ub828\ubd84\ud560<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uad6c\uac04 \\([a,b]\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(f\\colon[a,b]\\to\\mathbb R\\)\uacfc \\([a,b]\\)\uc758 \ubd84\ud560 \\(P\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\begin{aligned}
\nm_i&=\\inf\\{f(x)\\mid x_{i-1}\\le x\\le x_i\\},\\\\[3pt]
\nM_i&=\\sup\\{f(x)\\mid x_{i-1}\\le x\\le x_i\\}
\n\\end{aligned}\\]
\n\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(P\\)\uc5d0 \ub300\ud55c \\(f\\)\uc758 \ub9ac\ub9cc \uc0c1\ud569<\/span>(upper Riemann sum)\uacfc \ub9ac\ub9cc \ud558\ud569<\/span>(lower Riemann sum)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[U(f,P)=\\sum_{i=1}^nM_i(x_i-x_{i-1}),\\quad L(f,P)=\\sum_{i=1}^nm_i(x_i-x_{i-1}).\\]
\n\ud568\uc218 \\(f\\)\uc640 \uad6c\uac04 \\([a,b]\\)\uac00 \uace0\uc815\ub418\uc5b4 \uc788\uc744 \ub54c, \uc0c1\ud569\uc758 \uac12\uacfc \ud558\ud569\uc758 \uac12\uc740 \ubd84\ud560 \\(P\\)\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c8 \uc218 \uc788\ub2e4.<\/p>\n

\n

\ubb38\uc81c 6.1.<\/span>
\n\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,b]\\)\uc5d0\uc11c \uc720\uacc4\uc774\uace0, \\(P\\)\uc640 \\(Q\\)\uac00 \\([a,b]\\)\uc758 \ubd84\ud560\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n

    \n
  1. \\(L(f,P)\\le U(f,P)\\)<\/li>\n
  2. \\(Q\\)\uac00 \\(P\\)\uc758 \uc138\ub828\ubd84\ud560\uc77c \ub54c \\(L(f,P)\\le L(f,Q)\\)\uc774\uace0 \\(U(f,P)\\ge U(f,Q)\\)\uc774\ub2e4.<\/li>\n
  3. \\(L(f,P)\\le U(f,Q)\\)<\/li>\n<\/ol>\n<\/div>\n

    \ud568\uc218 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \uc720\uacc4\uc774\uace0 \\(P\\)\uac00 \\([a,b]\\)\uc758 \ubd84\ud560\uc77c \ub54c \\(P\\)\uc5d0 \ub300\ud55c \\(f\\)\uc758 \uc0c1\ud569\uacfc \ud558\ud569\uc740 \uac01\uac01 \uc720\uacc4\uc774\ub2e4. \ud2b9\ud788 \\([a,b]\\)\uc758 \uc784\uc758\uc758 \ubd84\ud560 \\(P\\), \\(Q\\)\uc5d0 \ub300\ud558\uc5ec \\(L(f,P)\\le U(f,Q)\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc0c1\uc801\ubd84<\/span>(upper integral)\uacfc \ud558\uc801\ubd84<\/span>(lower integral)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
    \n\\[\\begin{aligned}
    \n\\overline{\\int_a^b}f&=\\inf\\left\\{U(f,P)\\mid P\\text{\ub294 }[a,b]\\text{\uc758 \ubd84\ud560}\\right\\},\\\\
    \n\\underline{\\int_a^b}f&=\\sup\\left\\{L(f,P)\\mid P\\text{\ub294 }[a,b]\\text{\uc758 \ubd84\ud560}\\right\\}.
    \n\\end{aligned}\\]
    \n\ud568\uc218 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5<\/span>\ud558\ub2e4(Riemann integrable)\ub294 \uac83\uc740 \\([a,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \uc720\uacc4\uc774\uace0 \\(f\\)\uc758 \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc774 \uac19\uc740 \uac83\uc774\ub2e4. \uc774 \uacf5\ud1b5\uac12\uc744
    \n\\[\\int_a^b f(x)\\,dx\\]
    \n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774 \uac12\uc744 \uac04\ub2e8\ud788 \\(\\int_a^b f\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n

    \ub9ac\ub9cc \uc801\ubd84\uc758 \uc815\uc758\uc640 \uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 6.1. (\uc801\ubd84 \uac00\ub2a5\uc131\uc5d0 \ub300\ud55c \ub9ac\ub9cc \ud310\uc815\ubc95)<\/span><\/p>\n

    \ud568\uc218 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740, \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud574 \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(U(f,P)-L(f,P)<\\varepsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \uc801\ubd84\uac12\uc744 \\(I\\)\ub77c\uace0 \ud558\uc790. \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud574 \ubd84\ud560 \\(P\\), \\(Q\\)\uac00 \uc874\uc7ac\ud558\uc5ec
    \n\\[
    \nU(f,P)<I+\\frac{\\varepsilon}{2},\\quad
    \nL(f,Q)>I-\\frac{\\varepsilon}{2}
    \n\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \\(R=P\\cup Q\\)\ub294 \\(P\\)\uc640 \\(Q\\)\uc758 \uacf5\ud1b5\uc138\ub828\ubd84\ud560\uc774\ubbc0\ub85c
    \n\\[
    \nU(f,R)-L(f,R)
    \n\\le U(f,P)-L(f,Q)<\\varepsilon
    \n\\]
    \n\uc774\ub2e4.<\/p>\n

    \ubc18\ub300\ub85c \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud574 \\(U(f,P)-L(f,P)<\\varepsilon\\)\uc778 \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790. \ud558\uc801\ubd84\uacfc \uc0c1\uc801\ubd84\uc744 \uac01\uac01 \\(\\underline I\\), \\(\\overline I\\)\ub77c\uace0 \uc4f0\uba74 \ubaa8\ub4e0 \ubd84\ud560 \\(P\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \nL(f,P)\\le\\underline I\\le\\overline I\\le U(f,P)
    \n\\]
    \n\uc774\ubbc0\ub85c
    \n\\[
    \n0\\le\\overline I-\\underline I\\le U(f,P)-L(f,P)<\\varepsilon.
    \n\\]
    \n\\(\\varepsilon\\)\uc774 \uc784\uc758\uc801\uc774\ubbc0\ub85c \\(\\overline I=\\underline I\\)\uc774\uace0, \ub530\ub77c\uc11c \\(f\\)\ub294 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/span><\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 6.2.<\/span>
    \n\uc801\ubd84\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \uad6c\uac04 \\([0,1]\\)\uc5d0\uc11c \ud568\uc218 \\(f(x)=x^2\\)\uc758 \uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 6.3.<\/span>
    \n\ud568\uc218 \\(D\\colon[0,1]\\to\\mathbb R\\)\ub97c
    \n\\[
    \nD(x)=
    \n\\begin{cases}
    \n1 & \\text{if }\\;x\\in\\mathbb Q,\\\\[5pt]
    \n0 & \\text{if }\\;x\\notin\\mathbb Q
    \n\\end{cases}
    \n\\]
    \n\ub77c\uace0 \uc815\uc758\ud558\uc790. \uc774 \ud568\uc218\ub97c \ub514\ub9ac\ud074\ub808 \ud568\uc218<\/span>(Dirichlet function)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(D\\)\uac00 \\([0,1]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 6.4.<\/span>
    \n\ud568\uc218 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \ud568\uc218 \\(g\\)\uac00 \\([a,b]\\)\uc758 \uc720\ud55c \uac1c\uc758 \uc810\uc744 \uc81c\uc678\ud558\uba74 \\(f\\)\uc640 \uac19\ub2e4\uace0 \ud558\uc790. \\(g\\)\ub3c4 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uace0
    \n\\[
    \n\\int_a^b g(x)\\,dx=\\int_a^b f(x)\\,dx
    \n\\]
    \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

    \uc0c1\ud569\uacfc \ud558\ud569 \ub300\uc2e0 \ub9ac\ub9cc\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub9ac\ub9cc \uc801\ubd84\uc744 \uc815\uc758\ud560 \uc218\ub3c4 \uc788\ub2e4. \uac01 \uc18c\uad6c\uac04 \\([x_{i-1},x_i]\\)\uc5d0\uc11c \uc810 \\(t_i\\)\ub97c \uc120\ud0dd\ud558\uc790. \uc774 \uc810\uc744 \ud0dc\uadf8<\/span>(tag)\ub77c\uace0 \ubd80\ub978\ub2e4. \ud0dc\uadf8\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub9cc\ub4e0 \ud569
    \n\\[
    \nS(f,P,\\{t_i\\})=\\sum_{i=1}^n f(t_i)(x_i-x_{i-1})
    \n\\]
    \n\uc744 \\(P\\)\uc640 \\(\\{t_i\\}\\)\uc5d0 \ub300\ud55c \\(f\\)\uc758 \ub9ac\ub9cc \ud569<\/span>(Riemann sum)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\lVert P\\rVert\\to0\\)\uc77c \ub54c \\(f\\)\uc758 \ub9ac\ub9cc \ud569\uc774 \uac12 \\(I\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta>0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(\\lVert P\\rVert<\\delta\\)\uc778 \uc784\uc758\uc758 \ubd84\ud560 \\(P\\)\uc640 \\(P\\)\uc758 \uac01 \uc18c\uad6c\uac04\uc5d0\uc11c \ud0dd\ud55c \uc810\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc784\uc758\uc758 \uc720\ud55c\uc218\uc5f4 \\(\\{t_i\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(|S(f,P,\\{t_i\\})-I|<\\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \uc774\uac83\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.
    \n\\[
    \n\\lim_{\\lVert P\\rVert\\to0}S(f,P,\\{t_i\\})=I.\\tag{6.1}
    \n\\]<\/p>\n

    \ub9ac\ub9cc\uc740 \uc801\ubd84\uc744 \ub9ac\ub9cc\ud569\uc758 \uadf9\ud55c\uc73c\ub85c \uc815\uc758\ud588\ub2e4. \uc0c1\ud569\uacfc \ud558\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ud55c \uc801\ubd84\uc740 \ubcf8\ub798 \ub2e4\ub974\ubd80(Darboux)\uc758 \uc801\ubd84\uc774\ub2e4. \uc720\uacc4\uc778 \ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub294 \ub450 \uc815\uc758\uac00 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 6.2. (\ub9ac\ub9cc\ud569\uacfc \ub2e4\ub974\ubd80 \uc801\ubd84\uc758 \ub3d9\uce58)<\/span><\/p>\n

    \\([a,b]\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(f\\)\uc640 \uc2e4\uc218 \\(I\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ub450 \uc870\uac74\uc740 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.<\/p>\n

      \n
    1. \\(f\\)\ub294 \\([a,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \\(\\int_a^b f=I\\)\uc774\ub2e4.<\/li>\n
    2. \\(\\lVert P\\rVert\\to0\\)\uc77c \ub54c \ubaa8\ub4e0 \ud0dc\uadf8 \uc120\ud0dd\uc5d0 \ub300\ud558\uc5ec \\(S(f,P,\\{t_i\\})\\to I\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
      \n

      \uc99d\uba85<\/span>
      \n\uc5b4\ub5a4 \\(M>0\\)\uc5d0 \ub300\ud558\uc5ec \\(|f|\\le M\\)\uc774\ub77c\uace0 \ud558\uc790. \uba3c\uc800 \\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \uc801\ubd84\uac12\uc774 \\(I\\)\ub77c\uace0 \ud558\uc790. \\(\\varepsilon>0\\)\uc744 \uace0\uc815\ud558\uace0
      \n\\[
      \nU(f,P_0)-L(f,P_0)<\\frac{\\varepsilon}{2}
      \n\\]
      \n\uc778 \ubd84\ud560 \\(P_0=\\{x_0,\\ldots,x_m\\}\\)\uc744 \ud0dd\ud55c\ub2e4. \uc784\uc758\uc758 \ubd84\ud560 \\(P\\)\uc5d0 \\(P_0\\)\uc758 \ub0b4\ubd80 \ubd84\ud560\uc810\ub4e4\uc744 \ud558\ub098\uc529 \ucd94\uac00\ud558\uc5ec \uacf5\ud1b5\uc138\ub828\ubd84\ud560 \\(P’=P\\cup P_0\\)\uc744 \ub9cc\ub4e4\uc790. \ud55c \ubd84\ud560\uc810\uc744 \ucd94\uac00\ud560 \ub54c \uc0c1\ud569\uc774\ub098 \ud558\ud569\uc758 \ubcc0\ud654\ub7c9\uc758 \uc808\ub313\uac12\uc740 \\(2M\\lVert P\\rVert\\) \uc774\ud558\uc774\ubbc0\ub85c
      \n\\[
      \nU(f,P)\\le U(f,P’)+2Mm\\lVert P\\rVert,
      \n\\quad
      \nL(f,P)\\ge L(f,P’)-2Mm\\lVert P\\rVert.
      \n\\]
      \n\ub610\ud55c \\(P’\\)\ub294 \\(P_0\\)\uc758 \uc138\ub828\ubd84\ud560\uc774\ubbc0\ub85c
      \n\\[
      \nU(f,P’)\\le U(f,P_0),\\quad L(f,P’)\\ge L(f,P_0).
      \n\\]
      \n\ub530\ub77c\uc11c \\(P\\)\uc758 \uc784\uc758\uc758 \ud0dc\uadf8\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \nL(f,P_0)-2Mm\\lVert P\\rVert
      \n\\le S(f,P,\\{t_i\\})
      \n\\le U(f,P_0)+2Mm\\lVert P\\rVert.
      \n\\]
      \n\\(I\\)\ub3c4 \\(L(f,P_0)\\)\uc640 \\(U(f,P_0)\\) \uc0ac\uc774\uc5d0 \uc788\uc73c\ubbc0\ub85c \\(\\lVert P\\rVert<\\varepsilon\/(4Mm)\\)\uc774 \ub418\ub3c4\ub85d \ud558\uba74
      \n\\[
      \n|S(f,P,\\{t_i\\})-I|<\\varepsilon.
      \n\\]
      \n\uc989 \ub9ac\ub9cc\ud569\uc740 \ud0dc\uadf8 \uc120\ud0dd\uacfc \ubb34\uad00\ud558\uac8c \\(I\\)\ub85c \uc218\ub834\ud55c\ub2e4.<\/p>\n

      \ubc18\ub300\ub85c \ubaa8\ub4e0 \ub9ac\ub9cc\ud569\uc774 \\(I\\)\ub85c \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lVert P\\rVert<\\delta\\)\uc774\uba74 \ubaa8\ub4e0 \ud0dc\uadf8 \ub9ac\ub9cc\ud569\uc774 \\(I\\)\uc640 \\(\\varepsilon\/4\\)\ubcf4\ub2e4 \uc791\uc740 \uac70\ub9ac \uc548\uc5d0 \uc788\ub3c4\ub85d \\(\\delta>0\\)\uc744 \ud0dd\ud55c\ub2e4. \\(\\lVert P\\rVert<\\delta\\)\uc778 \ubd84\ud560 \\(P\\)\ub97c \ud558\ub098 \uace0\uc815\ud558\uc790. \uac01 \uc18c\uad6c\uac04\uc5d0\uc11c \uc0c1\ud55c\uacfc \ud558\ud55c\uc5d0 \uc784\uc758\ub85c \uac00\uae5d\uac8c \ud568\uc22b\uac12\uc744 \uac16\ub294 \ud0dc\uadf8\ub97c \uace0\ub974\uba74 \ub450 \ub9ac\ub9cc\ud569\uc758 \ucc28\ub97c \\(U(f,P)-L(f,P)\\)\uc5d0 \uc784\uc758\ub85c \uac00\uae5d\uac8c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \ud55c\ud3b8 \uc784\uc758\uc758 \ub450 \ud0dc\uadf8 \ub9ac\ub9cc\ud569 \\(S_1\\), \\(S_2\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \n|S_1-S_2|\\le|S_1-I|+|S_2-I|<\\frac{\\varepsilon}{2}.
      \n\\]
      \n\ub530\ub77c\uc11c \\(U(f,P)-L(f,P)\\le\\varepsilon\/2<\\varepsilon\\)\uc774\ub2e4. \uc815\ub9ac 6.1\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub2e4\uc2dc \ub9ac\ub9cc\ud569\uc758 \uc218\ub834\uac12\uc740 \uc801\ubd84\uac12\uacfc \uac19\uc73c\ubbc0\ub85c \\(\\int_a^b f=I\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

      \n

      \ubb38\uc81c 6.5.<\/span>
      \n\ud568\uc218 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \ud568\uc218 \\(g\\)\uac00 \\([c,d]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba70 \\(f([a,b])\\subseteq[c,d]\\)\uc77c \ub54c, \ud569\uc131\ud568\uc218 \\(g\\circ f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

      \n

      \ubb38\uc81c 6.6.<\/span>
      \n\ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uba74 \\(f+g\\)\uc640 \\(fg\\)\ub3c4 \\([a,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

      \n

      \ubb38\uc81c 6.7.<\/span>
      \n\ubd84\ud560 \\(a=x_0<x_1<\\cdots<x_n=b\\)\uc640 \uc2e4\uc218 \\(c_1,\\ldots,c_n\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uac01 \uc5f4\ub9b0\uad6c\uac04 \\((x_{i-1},x_i)\\)\uc5d0\uc11c \\(s(x)=c_i\\)\uc774\uace0 \ubd84\ud560\uc810\uc5d0\uc11c\uc758 \uac12\uc740 \uc784\uc758\ub85c \uc815\ud55c \ud568\uc218 \\(s\\)\ub97c \uc0dd\uac01\ud558\uc790. \\(s\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uace0
      \n\\[
      \n\\int_a^b s(x)\\,dx=\\sum_{i=1}^n c_i(x_i-x_{i-1})
      \n\\]
      \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

      \n

      \ubb38\uc81c 6.8.<\/span>
      \n\ud568\uc218 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc815\ub9ac 6.2\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.
      \n\\[
      \n\\int_a^b f(x)\\,dx
      \n=
      \n\\lim_{n\\to\\infty}\\frac{b-a}{n}\\sum_{k=1}^n f\\left(a+\\frac{b-a}{n}k\\right).
      \n\\]<\/p>\n<\/div>\n

      \uc801\ubd84 \uac00\ub2a5\uc131 \uc870\uac74<\/h3>\n

      \uc801\ubd84\uc744 \uc815\uc758\ud55c \ub4a4\uc5d0\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ub450 \uac00\uc9c0 \uc758\ubb38\uc774 \uc0dd\uae34\ub2e4.<\/p>\n

        \n
      • \uc5b4\ub5a0\ud55c \ud568\uc218\uac00 \uc801\ubd84 \uac00\ub2a5\ud55c\uac00?<\/li>\n
      • \uc801\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uac00 \uc788\uc744 \ub54c, \uadf8 \ud568\uc218\uc758 \uc801\ubd84\uac12\uc744 \uc5b4\ub5bb\uac8c \uad6c\ud558\ub294\uac00?<\/li>\n<\/ul>\n
        \n

        \uc815\ub9ac 6.3. (\uc5f0\uc18d\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131)<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \\(f\\)\ub294 \\([a,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/span>
        \n\\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \uc774 \uad6c\uac04\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \\(\\varepsilon>0\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uade0\ub4f1\uc5f0\uc18d\uc131\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta>0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(|x-y|<\\delta\\)\uc778 \uc784\uc758\uc758 \\(x,y\\in[a,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x)-f(y)|<\\varepsilon\/(b-a)\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \\(\\lVert P\\rVert<\\delta\\)\uc778 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud558\uba74 \\(M_i-m_i<\\varepsilon\/(b-a)\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(U(f,P)-L(f,P)<\\varepsilon\\)\uc774\uace0, \uc815\ub9ac 6.1\uc5d0 \uc758\ud574 \\(f\\)\ub294 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/span><\/p>\n<\/div>\n

        \n

        \uc815\ub9ac 6.4. (\ub2e8\uc870\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131)<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \ub2e8\uc870\uc774\uba74, \\(f\\)\ub294 \\([a,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/span>
        \n\ub9cc\uc57d \\(f(a)=f(b)\\)\uc774\uba74 \\(f\\)\ub294 \uc0c1\uc218\ud568\uc218\uc774\ubbc0\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \uc774\uc81c \\(f(a)\\ne f(b)\\)\ub77c\uace0 \ud558\uc790. \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790.<\/p>\n

        \\(\\varepsilon>0\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(\\lVert P\\rVert<\\varepsilon\/(f(b)-f(a))\\)\uc778 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud558\uba74
        \n\\[
        \nU(f,P)-L(f,P)
        \n\\le\\lVert P\\rVert(f(b)-f(a))
        \n<\\varepsilon.
        \n\\]
        \n\ub530\ub77c\uc11c \uc815\ub9ac 6.1\uc5d0 \uc758\ud574 \\(f\\)\ub294 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/span><\/p>\n<\/div>\n

        \ud568\uc218 \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc774 \uad6c\uac04 \\([a,b]\\)\uc5d0 \uc5b4\ub5bb\uac8c \ubd84\ud3ec\ud574 \uc788\ub294\uc9c0\uc5d0 \ub530\ub77c \\(f\\)\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud560 \uc218 \uc788\ub2e4. \uc774\uc640 \uad00\ub828\ub41c \uc815\ub9ac\ub97c \uc9c4\uc220\ud558\uae30 \uc704\ud574\uc11c\ub294 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc758 \uac1c\ub150\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n

        \uc9d1\ud569 \\(E\\subseteq\\mathbb R\\)\uc774 \ub974\ubca0\uadf8 \uce21\ub3c4 0<\/span>(Lebesgue measure zero)\uc778 \uc9d1\ud569\uc774\ub77c\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \uac00\uc0b0 \uac1c\uc758 \uc5f4\ub9b0\uad6c\uac04 \\(\\{I_n\\}\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(E\\subseteq\\bigcup_{n=1}^{\\infty}I_n\\)\uc774\uace0 \\(\\sum_{n=1}^{\\infty}|I_n|<\\varepsilon\\)\uc778 \uac83\uc774\ub2e4. \uc5ec\uae30\uc11c \\(|I_n|\\)\uc740 \uad6c\uac04 \\(I_n\\)\uc758 \uae38\uc774\ub97c \ub73b\ud55c\ub2e4.<\/p>\n

          \n
        • \uc720\ud55c\uc9d1\ud569\uc758 \uce21\ub3c4\ub294 \\(0\\)\uc774\ub2e4.<\/li>\n
        • \uac00\uc0b0\uc9d1\ud569\uc758 \uce21\ub3c4\ub294 \\(0\\)\uc774\ub2e4.<\/li>\n
        • \uce78\ud1a0\uc5b4\uc758 \uc9d1\ud569\uc740 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\uc9c0\ub9cc \uce21\ub3c4 \\(0\\)\uc774\ub2e4. \uc2e4\uc81c\ub85c \\(n\\)\ubc88\uc9f8 \ub2e8\uacc4\uc5d0 \ub0a8\ub294 \\(2^n\\)\uac1c\uc758 \uad6c\uac04\uc758 \uae38\uc774\uc758 \ud569\uc740 \\((2\/3)^n\\)\uc774\uace0, \uc774 \uad6c\uac04\ub4e4\uc744 \uc870\uae08 \ub113\ud78c \uc5f4\ub9b0\uad6c\uac04\ub4e4\ub85c\ub3c4 \ucd1d\uae38\uc774\ub97c \uc784\uc758\ub85c \uc791\uac8c \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/li>\n
        • \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04 \\([a,b]\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\uace0, \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4.<\/li>\n<\/ul>\n
          \n

          \ubb38\uc81c 6.9.<\/span>
          \n\uac00\uc0b0\uc9d1\ud569\uc774 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

          \n

          \ubb38\uc81c 6.10.<\/span>
          \n\uac00\uc0b0 \uac1c\uc758 \uc9d1\ud569 \\(A_1,A_2,A_3,\\ldots\\)\uac00 \ubaa8\ub450 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc774\uba74, \ud569\uc9d1\ud569
          \n\\[
          \nA_1\\cup A_2\\cup A_3\\cup\\cdots
          \n\\]
          \n\ub3c4 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

          \n

          \ubb38\uc81c 6.11.<\/span>
          \n\uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

          \n

          \uc815\ub9ac 6.5. (\uc801\ubd84 \uac00\ub2a5\uc131\uc5d0 \ub300\ud55c \ub974\ubca0\uadf8 \uc815\ub9ac)<\/span><\/p>\n

          \uad6c\uac04 \\([a,b]\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(f\\)\uac00 \uc774 \uad6c\uac04\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\([a,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uc9d1\ud569\uc774 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/span>
          \n\uc810 \\(x\\in[a,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ud568\uc218\uc758 \uc9c4\ub3d9<\/span>(oscillation)\uc744
          \n\\[
          \n\\omega_f(x)
          \n=
          \n\\inf_{\\delta>0}
          \n\\sup\\left\\{|f(y)-f(z)|\\;\\middle|\\;y,z\\in[a,b],\\ |y-x|<\\delta,\\ |z-x|<\\delta\\right\\}
          \n\\]
          \n\ub85c \uc815\uc758\ud55c\ub2e4. \\(f\\)\uac00 \\(x\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\omega_f(x)=0\\)\uc774\ub2e4. \ub610\ud55c \\(\\alpha>0\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[
          \nD_\\alpha=\\{x\\in[a,b]\\mid\\omega_f(x)\\ge\\alpha\\}
          \n\\]
          \n\ub77c\uace0 \ub193\uc73c\uba74 \\(D_\\alpha\\)\ub294 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4. \uc2e4\uc81c\ub85c \\(\\omega_f(x)<\\alpha\\)\uc774\uba74 \uc5b4\ub5a4 \\(r>0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\((x-2r,x+2r)\\cap[a,b]\\)\uc5d0\uc11c\uc758 \uc9c4\ub3d9\uc774 \\(\\alpha\\)\ubcf4\ub2e4 \uc791\uace0, \ub530\ub77c\uc11c \\(|y-x|<r\\)\uc778 \ubaa8\ub4e0 \\(y\\)\uc5d0 \ub300\ud558\uc5ec \\(\\omega_f(y)<\\alpha\\)\uc774\ub2e4.<\/p>\n

          \uba3c\uc800 \\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ubd88\uc5f0\uc18d\uc810\uc758 \uc9d1\ud569\uc740
          \n\\[
          \nD=\\bigcup_{n=1}^{\\infty}D_{1\/n}
          \n\\]
          \n\uc774\ub2e4. \\(n\\in\\mathbb N\\)\uacfc \\(\\varepsilon>0\\)\uc744 \uace0\uc815\ud558\uc790. \ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec
          \n\\[
          \nU(f,P)-L(f,P)<\\frac{\\varepsilon}{2n}
          \n\\]
          \n\uc778 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud560 \uc218 \uc788\ub2e4. \\(P\\)\uc758 \ubd84\ud560\uc810\uc774 \uc544\ub2cc \\(x\\in D_{1\/n}\\)\uc774 \uc5b4\ub5a4 \uc18c\uad6c\uac04\uc758 \ub0b4\ubd80\uc5d0 \uc18d\ud558\uba74 \uadf8 \uc18c\uad6c\uac04\uc5d0\uc11c\uc758 \\(f\\)\uc758 \uc9c4\ub3d9\uc740 \uc801\uc5b4\ub3c4 \\(1\/n\\)\uc774\ub2e4. \ub530\ub77c\uc11c \uc774\ub7ec\ud55c \uc18c\uad6c\uac04\ub4e4\uc758 \uae38\uc774\uc758 \ud569\uc740
          \n\\[
          \nn\\bigl(U(f,P)-L(f,P)\\bigr)<\\frac{\\varepsilon}{2}
          \n\\]
          \n\ubcf4\ub2e4 \uc791\ub2e4. \uc774 \uc720\ud55c \uac1c\uc758 \uc18c\uad6c\uac04\uc744 \uc870\uae08\uc529 \ub113\ud78c \uc5f4\ub9b0\uad6c\uac04\ub4e4\uacfc, \\(P\\)\uc758 \uc720\ud55c \uac1c\uc758 \ubd84\ud560\uc810\uc744 \ub36e\ub294 \uae38\uc774\uc758 \ud569\uc774 \\(\\varepsilon\/2\\)\ubcf4\ub2e4 \uc791\uc740 \uc5f4\ub9b0\uad6c\uac04\ub4e4\uc744 \ud568\uaed8 \ud0dd\ud558\uba74 \\(D_{1\/n}\\)\uc744 \uae38\uc774\uc758 \ud569\uc774 \\(\\varepsilon\\)\ubcf4\ub2e4 \uc791\uc740 \uc5f4\ub9b0\uad6c\uac04\ub4e4\ub85c \ub36e\uc744 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \uac01 \\(D_{1\/n}\\)\uc740 \uce21\ub3c4 \\(0\\)\uc774\ub2e4. \ubb38\uc81c 6.10\uc758 \uacb0\uacfc\uc5d0 \uc758\ud558\uc5ec \\(D\\)\ub3c4 \uce21\ub3c4 \\(0\\)\uc774\ub2e4.<\/p>\n

          \ubc18\ub300\ub85c \ubd88\uc5f0\uc18d\uc810\uc758 \uc9d1\ud569 \\(D\\)\uac00 \uce21\ub3c4 \\(0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(|f|\\le M\\)\uc774\ub77c\uace0 \ud558\uc790. \\(M=0\\)\uc774\uba74 \uc790\uba85\ud558\ubbc0\ub85c \\(M>0\\)\uc774\ub77c\uace0 \uac00\uc815\ud55c\ub2e4. \\(\\varepsilon>0\\)\uc744 \uace0\uc815\ud558\uace0
          \n\\[
          \n\\eta=\\frac{\\varepsilon}{2(b-a)}
          \n\\]
          \n\ub77c\uace0 \ub193\uc790. \\(D_\\eta\\)\ub294 \ub2eb\ud78c\uc9d1\ud569\uc774\ubbc0\ub85c \\([a,b]\\)\uc758 \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569\uc774\ub2e4. \ub610\ud55c \\(D_\\eta\\subseteq D\\)\uc774\ubbc0\ub85c \uce21\ub3c4 \\(0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(D_\\eta\\)\ub97c \ub36e\ub294 \uc5f4\ub9b0\uad6c\uac04\ub4e4 \uc911 \uc720\ud55c \uac1c \\(I_1,\\ldots,I_r\\)\ub97c \ud0dd\ud558\uc5ec
          \n\\[
          \nD_\\eta\\subseteq G:=\\bigcup_{j=1}^r I_j,
          \n\\quad
          \n\\sum_{j=1}^r|I_j|<\\frac{\\varepsilon}{4M}
          \n\\]
          \n\uc774 \ub418\uac8c \ud560 \uc218 \uc788\ub2e4.<\/p>\n

          \\(K=[a,b]\\setminus G\\)\ub77c\uace0 \ud558\uc790. \\(K=\\varnothing\\)\uc774\uba74 \\([a,b]\\subseteq G\\)\uc774\ubbc0\ub85c \\(b-a\\le\\sum_{j=1}^r|I_j|\\)\uc774\uace0, \uc784\uc758\uc758 \ubd84\ud560 \\(P\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[
          \nU(f,P)-L(f,P)\\le2M(b-a)<\\frac{\\varepsilon}{2}
          \n\\]
          \n\uc774\ubbc0\ub85c \ubc14\ub85c \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c \\(K\\ne\\varnothing\\)\uc774\ub77c\uace0 \ud558\uc790. \uac01 \\(x\\in K\\)\uc5d0 \ub300\ud558\uc5ec \\(\\omega_f(x)<\\eta\\)\uc774\ubbc0\ub85c \uc5b4\ub5a4 \\(r_x>0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\((x-2r_x,x+2r_x)\\cap[a,b]\\)\uc5d0\uc11c\uc758 \uc9c4\ub3d9\uc774 \\(\\eta\\)\ubcf4\ub2e4 \uc791\ub2e4. \\(K\\)\uc758 \ucef4\ud329\ud2b8\uc131\uc5d0 \uc758\ud574 \\(K\\)\ub294 \uc720\ud55c \uac1c\uc758 \uad6c\uac04 \\((x_j-r_{x_j},x_j+r_{x_j})\\)\ub85c \ub36e\uc778\ub2e4. \uc774 \uc720\ud55c \uac1c\uc758 \\(r_{x_j}\\)\uc758 \ucd5c\uc19f\uac12\uc744 \\(\\delta>0\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

          \uc774\uc81c \\(I_1,\\ldots,I_r\\)\uc758 \ub05d\uc810\ub4e4\uc744 \ubd84\ud560\uc810\uc73c\ub85c \ud3ec\ud568\ud558\uace0 \ub178\ub984\uc774 \\(\\delta\\)\ubcf4\ub2e4 \uc791\uc740 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud55c\ub2e4. \\(P\\)\uc758 \uac01 \uc18c\uad6c\uac04\uc740 \\(G\\)\uc5d0 \uc644\uc804\ud788 \ud3ec\ud568\ub418\uac70\ub098 \\(K\\)\uc640 \ub9cc\ub09c\ub2e4. \\(K\\)\uc640 \ub9cc\ub098\ub294 \uc18c\uad6c\uac04\uc758 \uc9c4\ub3d9\uc740 \\(\\eta\\)\ubcf4\ub2e4 \uc791\uace0, \\(G\\)\uc5d0 \ud3ec\ud568\ub418\ub294 \uc18c\uad6c\uac04\uc758 \uc9c4\ub3d9\uc740 \\(2M\\) \uc774\ud558\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
          \n\\[
          \nU(f,P)-L(f,P)
          \n<\\eta(b-a)+2M\\sum_{j=1}^r|I_j|
          \n<\\frac{\\varepsilon}{2}+\\frac{\\varepsilon}{2}
          \n=\\varepsilon.
          \n\\]
          \n\ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/span><\/p>\n<\/div>\n

          \ub974\ubca0\uadf8 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \uc27d\uac8c \ud310\ubcc4\ud560 \uc218 \uc788\ub294 \ud568\uc218\uc758 \uc608\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n