{"id":3665,"date":"2019-05-02T01:05:17","date_gmt":"2019-05-01T16:05:17","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=3665"},"modified":"2021-12-14T19:04:52","modified_gmt":"2021-12-14T10:04:52","slug":"calculus-lebesgue-theorem-for-riemann-integrability","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-lebesgue-theorem-for-riemann-integrability\/","title":{"rendered":"\uc801\ubd84 \uac00\ub2a5\uc131\uc5d0 \ub300\ud55c \ub974\ubca0\uadf8\uc758 \uc815\ub9ac"},"content":{"rendered":"
\n\uc774 \ud3ec\uc2a4\ud2b8\ub294 \ubbf8\uc801\ubd84\ud559\ubcf4\ub2e4 \uc0c1\uae09 \uacfc\uc815\uc758 \ub0b4\uc6a9\uc744 \ub2e4\ub8e8\uace0 \uc788\uc2b5\ub2c8\ub2e4. \ubbf8\uc801\ubd84\ud559\uc744 \ucc98\uc74c \uacf5\ubd80\ud558\ub294 \ud559\uc0dd\ub4e4\uc740 \uc774 \ud3ec\uc2a4\ud2b8\uc758 \ub0b4\uc6a9\uc744 \uc774\ud574\ud558\uae30 \uc5b4\ub824\uc6b8 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc758 \ub0b4\uc6a9\uc744 \uc774\ud574\ud558\uae30 \uc704\ud574\uc11c\ub294 \ub9ac\ub9cc \uc801\ubd84\uc758 \uc5c4\ubc00\ud55c \uc815\uc758, \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uc5d0 \ub300\ud55c \ub9ac\ub9cc \ud310\uc815\ubc95, \uc0c1\ud55c\uacfc \ud558\ud55c\uc758 \uc131\uc9c8\uc744 \uc54c\uc544\uc57c \ud569\ub2c8\ub2e4. \ubbf8\uc801\ubd84\ud559\uc744 \ucc98\uc74c \uacf5\ubd80\ud558\uc9c0\ub9cc \uc774 \ud3ec\uc2a4\ud2b8\uc758 \ub0b4\uc6a9\uc744 \uaf2d \uc54c\uace0 \uc2f6\uc740 \uc0ac\ub78c\uc740 \uc815\uc758 1, \uc815\ub9ac 1, \uc608\uc81c 1, \uc815\ub9ac 2\uc758 \ub0b4\uc6a9(\ud480\uc774\uc640 \uc99d\uba85 \uc81c\uc678)\uacfc \uc608\uc81c 5, \uc608\uc81c 6\uc744 \ubcf4\uae30 \ubc14\ub78d\ub2c8\ub2e4.\n<\/div>\n

\ud2b9\uc815\ud55c \uad6c\uac04\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud560 \ub54c\uc5d0\ub294 \ub9ac\ub9cc \uc801\ubd84\uc758 \uc815\uc758\ub97c \uc774\uc6a9\ud558\uae30\ub3c4 \ud558\uace0 \ub9ac\ub9cc \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uae30\ub3c4 \ud55c\ub2e4. \uadf8\ub7ec\ub098 \ud568\uc218\uc758 \ubd88\uc5f0\uc18d\uc810\uc774 \ubd84\ud3ec\ud55c \ud615\ud0dc\ub97c \uad00\ucc30\ud568\uc73c\ub85c\uc368 \uad6c\uac04\uc758 \ubd84\ud560\uc744 \uc0dd\uac01\ud558\uc9c0 \uc54a\uace0\uc11c\ub3c4 \ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud560 \uc218 \uc788\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubd88\uc5f0\uc18d\uc810\uc758 \ubd84\ud3ec \ud615\ud0dc\ub97c \uc870\uc0ac\ud558\uc5ec \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\ubcc4\ub2e4\ub978 \uc5b8\uae09\uc774 \uc5c6\ub294 \ud55c \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c \ub2eb\ud78c \uad6c\uac04 \\([a,\\,b]\\)\uc640 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\ub294 \ud56d\uc0c1 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc744 \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n

<\/p>\n

\ub3c4\uc785<\/h3>\n

\\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \uc774 \uad6c\uac04\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218\uc774\uba70, \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\uac00 \uc720\ud55c\uc77c \ub54c \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \uadf8\ub7ec\ub098 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\uac00 \ubb34\ud55c\uc77c \uc9c0\ub77c\ub3c4 \\(f\\)\uac00 \uc774 \uad6c\uac04\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud560 \uc218\ub3c4 \uc788\ub2e4. \uc608\ub97c \ub4e4\uba74 \\(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\uadf8\ub7ec\uba74 \\(f\\)\ub294 \\([0,\\,1]\\)\uc5d0\uc11c \ubb34\ud55c\ud788 \ub9ce\uc740 \ubd88\uc5f0\uc18d\uc810\uc744 \uac16\uc9c0\ub9cc \uc774 \uad6c\uac04\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ud55c\ud3b8
\n\\[g(x) = \\begin{cases}
\n1 \\quad & \\text{if} \\,\\, x\\in \\mathbb{Q} \\\\[6pt]
\n0 \\quad & \\text{if} \\,\\, x\\notin \\mathbb{Q}
\n\\end{cases}\\]
\n\ub77c\uace0 \ud558\uba74 \\(g\\)\ub294 \\([0,\\,1]\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\uba70, \uc774 \uad6c\uac04\uc5d0\uc11c \uc801\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4.<\/p>\n

\uc5ec\uae30\uae4c\uc9c0\ub9cc \ubcf4\uba74 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \ubaa8\uc784\uc774 \uac00\uc0b0\uc778 \uacbd\uc6b0\ub294 \uc801\ubd84 \uac00\ub2a5\ud558\uace0, \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \ubaa8\uc784\uc774 \ube44\uac00\uc0b0\uc778 \uacbd\uc6b0\uc5d0\ub294 \uc801\ubd84 \ubd88\uac00\ub2a5\ud560 \uac83\ucc98\ub7fc \ubcf4\uc778\ub2e4. \uadf8\ub7ec\ub098 \uc774\uac83\uc740 \uc0ac\uc2e4\uc774 \uc544\ub2c8\ub2e4. \\(C\\)\uac00 \uce78\ud1a0\uc5b4 \uc9d1\ud569<\/a>(Cantor set)\uc774\uace0
\n\\[h(x) = \\begin{cases}
\n1 \\quad & \\text{if} \\,\\, x\\in C \\\\[6pt]
\n0 \\quad & \\text{if} \\,\\, x\\notin C
\n\\end{cases}\\]
\n\ub77c\uace0 \ud558\uba74 \\(h\\)\ub294 \\(C\\) \uc704\uc5d0\uc11c\ub9cc \ubd88\uc5f0\uc18d\uc774\ub2e4. \uc989 \\([0,\\,1]\\)\uc5d0\uc11c \\(h\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \ubaa8\uc784\uc740 \ube44\uac00\uc0b0\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(h\\)\ub294 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n

\ubd88\uc5f0\uc18d\uc810\uc758 \ubd84\ud3ec\ub97c \uc774\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud558\uae30 \uc704\ud574\uc11c\ub294 \uc9d1\ud569\uc744 \uac00\uc0b0\uc9d1\ud569\uacfc \ube44\uac00\uc0b0\uc9d1\ud569\uc73c\ub85c \uad6c\ubd84\ud558\ub294 \uac83\ub9cc\uc73c\ub85c\ub294 \ucda9\ubd84\ud558\uc9c0 \uc54a\uc73c\uba70, \ub2e4\ub978 \uad6c\ubd84\ubc95\uc744 \ub3c4\uc785\ud574\uc57c \ud55c\ub2e4.<\/p>\n

<\/p>\n

\uce21\ub3c4 0<\/h3>\n

\uad6c\uac04 \\([0,\\,1]\\)\uc740 \uae38\uc774\uac00 \\(1\\)\uc778 \uc9d1\ud569\uc774\ub2e4. \ubc18\uba74 \uc9d1\ud569 \\(\\left\\{ 1\/k \\,\\vert\\, k\\in\\mathbb{N}\\right\\}\\)\uc740 \uae38\uc774\uac00 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub2e4. \ub610\ud55c \uce78\ud1a0\uc5b4 \uc9d1\ud569 \\(C\\)\ub294 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uae38\uc774\uc758 \ud569\uc774 \\(\\epsilon\\)\ubcf4\ub2e4 \uc791\uc740 \uc5f4\ub9b0 \uad6c\uac04\ub4e4\ub85c \\(C\\)\ub97c \ub36e\uc744 \uc218 \uc788\uc73c\ubbc0\ub85c \uae38\uc774\uac00 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub77c \ud560 \uc218 \uc788\ub2e4. \uae38\uc774\uac00 \\(0\\)\uc778 \uc9d1\ud569\uc744 \uc5c4\ubc00\ud558\uac8c \uc815\uc758\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

\n

\uc815\uc758 1. (\uce21\ub3c4 0)<\/span><\/p>\n

\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(E\\)\uac00 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569<\/span>\uc774\ub77c\ub294 \uac83\uc740 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc5f4\ub9b0 \uad6c\uac04\ub4e4\uc758 \uac00\uc0b0\uc9d1\ud569 \\(\\left\\{ I_j \\right\\}\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(I_j\\)\uc758 \uae38\uc774\uc758 \ud569\uc774 \\(\\epsilon\\)\ubcf4\ub2e4 \uc791\uc73c\uba74\uc11c \\(\\left\\{ I_j \\right\\}\\)\uac00 \\(E\\)\ub97c \ub36e\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n<\/div>\n

\uc9c1\uad00\uc801\uc73c\ub85c, \uc9d1\ud569\uc758 \uce21\ub3c4\uac00 \\(0\\)\uc774\ub77c\ub294 \uac83\uc740 \ubb34\uc2dc\ud560 \uc218 \uc788\uc744 \uc815\ub3c4\ub85c \uc791\uc740 \uae38\uc774\ub97c \uac00\uc84c\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc0ac\uc2e4 \uce21\ub3c4\uac00 \\(0\\)\uc778 \uc9d1\ud569 \uc678\uc5d0\ub3c4 \uc591\uc218 \uce21\ub3c4(positive measure)\ub97c \uc815\uc758\ud560 \uc218 \uc788\uc9c0\ub9cc, \uce21\ub3c4\ub860\uc744 \uae4a\uc774 \uc0b4\ud3b4\ubcf4\ub294 \uac83\uc740 \uc6b0\ub9ac\uc758 \ubaa9\uc801\uc774 \uc544\ub2c8\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \uce21\ub3c4\uac00 \\(0\\)\uc778 \uacbd\uc6b0\ub9cc \uc815\uc758\ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 1. (\uce21\ub3c4 0\uc778 \uc9d1\ud569\uc758 \uc131\uc9c8)<\/span><\/p>\n

\uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uacfc \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \n
  1. \uce21\ub3c4\uac00 \\(0\\)\uc778 \uc9d1\ud569\uc758 \ubd80\ubd84\uc9d1\ud569\uc758 \uce21\ub3c4\ub294 \\(0\\)\uc774\ub2e4.<\/li>\n
  2. \uac00\uc0b0\uc9d1\ud569\uc758 \uce21\ub3c4\ub294 \\(0\\)\uc774\ub2e4.<\/li>\n
  3. \\(\\left\\{ E_k \\,\\vert\\, k\\in\\mathbb{N}\\right\\}\\)\uc774 \uce21\ub3c4\uac00 \\(0\\)\uc778 \uc9d1\ud569\uc758 \ubaa8\uc784\uc774\uba74 \\(E=\\bigcup_{k=1}^{\\infty} E_k\\)\uc758 \uce21\ub3c4\ub3c4 \\(0\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
    \n

    \uc99d\uba85<\/p>\n

    [1] \\(E\\)\uac00 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\uace0 \\(F\\subseteq E\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(E\\)\uac00 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ubbc0\ub85c \\(E\\)\ub97c \ub36e\uc73c\uba74\uc11c \uae38\uc774\uc758 \ud569\uc774 \\(\\epsilon\\) \ubbf8\ub9cc\uc778 \uad6c\uac04\ub4e4\uc758 \uac00\uc0b0\uc9d1\ud569 \\(\\left\\{ I_j \\right\\}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(\\left\\{ I_j \\right\\}\\)\ub294 \\(F\\)\ub3c4 \ub36e\uc73c\ubbc0\ub85c \\(F\\)\ub294 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub2e4.<\/p>\n

    [2] \\(E\\)\uac00 \uac00\ubd80\ubc88\uc9d1\ud569(\ubb34\ud55c\uc778 \uac00\uc0b0\uc9d1\ud569)\uc774\ub77c\uace0 \ud558\uc790. \uc989 \\[E= \\left\\{ x_j \\,\\vert\\, j\\in\\mathbb{N} \\right\\}\\]\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\[I_j = \\left( x_j – \\frac{\\epsilon}{2^{j+2}} ,\\, x_j + \\frac{\\epsilon}{2^{j+2}} \\right)\\]
    \n\uc774\ub77c\uace0 \ud558\uba74 \\(I_j\\)\uc758 \uae38\uc774\ub294 \\(2^{-j-1}\\epsilon\\)\uc774\ubbc0\ub85c \\(I_j\\)\ub4e4\uc758 \uae38\uc774\uc758 \ud569\uc740 \\(\\epsilon \/ 2\\)\uc774\ub2e4. \ub610\ud55c \uac01 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(x_j \\in I_j\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\left\\{ I_j \\right\\}\\)\ub294 \\(E\\)\ub97c \ub36e\uc73c\uba74\uc11c \uae38\uc774\uc758 \ud569\uc774 \\(\\epsilon\\) \ubbf8\ub9cc\uc778 \uc5f4\ub9b0 \uad6c\uac04\ub4e4\uc758 \ubaa8\uc784\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(E\\)\ub294 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub2e4. \ud55c\ud3b8 \uc720\ud55c\uc9d1\ud569\uc740 \uac00\ubd80\ubc88\uc9d1\ud569\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ubbc0\ub85c [1]\uc5d0 \uc758\ud558\uc5ec \uc720\ud55c\uc9d1\ud569\uc758 \uce21\ub3c4\ub3c4 \\(0\\)\uc774\ub2e4.<\/p>\n

    [3] \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uac01 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \uae38\uc774\uc758 \ud569\uc774 \\(2^{-k-1}\\epsilon\\) \ubbf8\ub9cc\uc774\uba74\uc11c \\(E_k\\)\ub97c \ub36e\ub294 \uc5f4\ub9b0 \uad6c\uac04\ub4e4\uc758 \ubaa8\uc784 \\(\\left\\{ I_{k,j} \\right\\}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\[\\left\\{ I_{k,j} \\,\\vert\\, k\\in\\mathbb{N} ,\\, j\\in\\mathbb{N} \\right\\}\\]\uc740 \uae38\uc774\uc758 \ud569\uc774 \\(\\epsilon \/ 2\\) \uc774\ud558\uc774\uace0 \\(E\\)\ub97c \ub36e\ub294 \uc5f4\ub9b0 \uad6c\uac04\ub4e4\uc758 \ubaa8\uc784\uc774\ubbc0\ub85c \\(E\\)\ub294 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub2e4.<\/p>\n

    <\/span><\/p>\n<\/div>\n

    \n

    \uc608\uc81c 1. (\uad6c\uac04\uc758 \uce21\ub3c4)<\/span><\/p>\n

    \ub2eb\ud78c \uad6c\uac04 \\([0,\\,1]\\)\uc758 \uce21\ub3c4\uac00 \\(0\\)\uc774 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624.\n<\/p>\n

    \ud480\uc774.<\/span>
    \n\uacb0\ub860\uacfc \ubc18\ub300\ub85c \\([0,\\,1]\\)\uc774 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon = 1\/2\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\([0,\\,1]\\)\uc744 \ub36e\uc73c\uba74\uc11c \uae38\uc774\uc758 \ud569\uc774 \\(\\epsilon\\) \ubbf8\ub9cc\uc778 \uc5f4\ub9b0 \uad6c\uac04\ub4e4\uc758 \ubaa8\uc784 \\(\\left\\{ I_j \\right\\}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \ud558\uc774\ub124-\ubcf4\ub810 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ I_j \\right\\}\\)\uc5d0\uc11c \uc720\ud55c \uac1c\uc758 \uc6d0\uc18c
    \n\\[(a_1 ,\\, b_1 ) ,\\,\\, (a_2 ,\\, b_2 ) ,\\,\\, \\cdots ,\\,\\, (a_n ,\\, b_n )\\]
    \n\uc744 \ud0dd\ud558\uc5ec \uc774\ub4e4\uc758 \ud569\uc9d1\ud569\uc774 \\([0,\\,1]\\)\uc744 \ub36e\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4. \uc774 \uad6c\uac04\ub4e4\uc758 \uac1c\uc218\ub294 \uc720\ud55c\uc774\ubbc0\ub85c
    \n\\[\\begin{align}
    \n0 & \\in (a_{r_1} ,\\, b_{r_1} ),\\\\[7pt]
    \nb_{r_1} & \\in (a_{r_2} ,\\, b_{r_2} ),\\\\[7pt]
    \nb_{r_2} & \\in (a_{r_3} ,\\, b_{r_3} ),\\\\[7pt]
    \n&\\,\\,\\vdots \\\\[7pt]
    \nb_{r_{n-1}} & \\in (a_{r_n} ,\\, b_{r_n} )
    \n\\end{align}\\]
    \n\uc774 \uc131\ub9bd\ud558\ub3c4\ub85d
    \n\\[(a_{r_1} ,\\, b_{r_1} ),\\,\\,(a_{r_2} ,\\, b_{r_2} ),\\,\\, \\cdots ,\\,\\,(a_{r_n} ,\\, b_{r_n} )\\]
    \n\uacfc \uac19\uc774 \uc21c\uc11c\ub97c \ubc14\uafb8\uc5b4 \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\uba74
    \n\\[[0,\\,1]\\subseteq (a_{r_1} ,\\, b_{r_1} ) \\cup (a_{r_2} ,\\, b_{r_2} ) \\cup \\cdots \\cup (a_{r_n} ,\\, b_{r_n} ) = (a_{r_1} ,\\, b_{r_n} )\\]
    \n\uc774\ubbc0\ub85c
    \n\\[a_{r_1} < 0 \\quad \\text{and} \\quad 1 < b_{r_n}\\]\n\uc989\n\\[b_{r_n} - a_{r_1} > 1\\tag{1}\\]
    \n\uc774\ub2e4. \uadf8\ub7f0\ub370
    \n\\[a_{r_2} < b_{r_1} ,\\,\\, a_{r_3} < b_{r_2} ,\\,\\, \\cdots ,\\,\\, a_{r_n} < b_{r_{n-1}}\\]\n\uc774\ubbc0\ub85c\n\\[\\begin{align}\nb_{r_n} - a_{r_1} &< ( b_{r_n} - a_{r_n} ) + (b_{r_{n-1}} - a_{r_{n-1}} ) + \\cdots \\\\[4pt]\n&\\quad + (b_{r_2} - a_{r_2}) + (b_{r_1} - a_{r_1} ) < \\epsilon = \\frac{1}{2} \\tag{2}\n\\end{align}\\]\n\uc774\ub2e4. (1)\uacfc (2)\ub294 \uc11c\ub85c \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\([0,\\,1]\\)\uc740 \uae38\uc774\uc758 \ud569\uc774 \\(\\epsilon\\)\ubcf4\ub2e4 \uc791\uc740 \uc5f4\ub9b0 \uad6c\uac04\ub4e4\ub85c \ub36e\uc744 \uc218 \uc5c6\ub2e4.\n<\/span><\/p>\n<\/div>\n

    \uc608\uc81c 1\uc758 \ud480\uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c, \uae38\uc774\uac00 \uc591\uc218\uc778 \ubaa8\ub4e0 \uad6c\uac04\uc740 \uce21\ub3c4\uac00 \\(0\\)\uc774 \uc544\ub2d8\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.<\/p>\n

    \n

    \uc608\uc81c 2. (\uce78\ud1a0\uc5b4 \uc9d1\ud569\uc758 \uce21\ub3c4)<\/span><\/p>\n

    \uce78\ud1a0\uc5b4 \uc9d1\ud569\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub9cc\ub4e4\uc5b4\uc9c4 \uc9d1\ud569\uc774\ub2e4. \uba3c\uc800 \uad6c\uac04 \\([0,\\,1]\\)\uc758 \uac00\uc6b4\ub370 \\(1\/3\\)\uc744 \uc81c\uac70\ud55c\ub2e4. \uc989
    \n\\[C_1 = \\left[ 0 ,\\, \\frac{1}{3} \\right] \\cup \\left[ \\frac{2}{3} ,\\, 1 \\right]\\]
    \n\ub2e4\uc74c\uc73c\ub85c \ub0a8\uc740 \ub450 \uad6c\uac04\uc758 \uac00\uc6b4\ub370 \\(1\/3\\)\uc744 \uc81c\uac70\ud55c\ub2e4. \uc989
    \n\\[C_2 = \\left[ 0,\\, \\frac{1}{9} \\right] \\cup \\left[ \\frac{2}{9} ,\\, \\frac{3}{9} \\right] \\cup \\left[ \\frac{6}{9} ,\\, \\frac{7}{9} \\right] \\cup \\left[ \\frac{8}{9} ,\\, 1 \\right]\\]
    \n\uc774\uc640 \uac19\uc774 \ub0a8\uc740 \uad6c\uac04\uc758 \uac00\uc6b4\ub370 \\(1\/3\\)\uc744 \uc81c\uac70\ud558\ub294 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec \ub2eb\ud78c \uc9d1\ud569\uc758 \uc5f4 \\(\\left\\{ C_i \\right\\}\\)\ub97c \uc5bb\ub294\ub2e4. \uc774\ub54c
    \n\\[C = \\bigcap_{i=1}^{\\infty} C_i\\]
    \n\ub97c \uce78\ud1a0\uc5b4 \uc9d1\ud569<\/span>(Cantor set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

      \n
    1. \uce78\ud1a0\uc5b4 \uc9d1\ud569\uc758 \uce21\ub3c4\uac00 \\(0\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
    2. \uce78\ud1a0\uc5b4 \uc9d1\ud569\uc774 \ube44\uac00\uc0b0\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n

      \ud480\uc774.<\/span>
      \n[1] \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c \\(C_n\\)\uc740 \uae38\uc774\uac00 \\(1\/3^n\\)\uc778 \\(2^n\\)\uac1c\uc758 \ub2eb\ud78c \uad6c\uac04\uc758 \ud569\uc9d1\ud569\uc774\ub2e4. \ub530\ub77c\uc11c \uc774 \ub2eb\ud78c \uad6c\uac04\ub4e4\uc758 \uae38\uc774\uc758 \ud569\uc740 \\((2\/3)^n\\)\uc774\ub2e4.<\/p>\n

      \\((2\/3)^N < \\epsilon \/ 2\\)\uc778 \uc790\uc5f0\uc218 \\(N\\)\uc744 \ud0dd\ud558\uc790. \uadf8\ub7ec\uba74 \\(C_N\\)\uc740 \\(2^N\\)\uac1c\uc758 \ub2eb\ud78c \uad6c\uac04\ub4e4\uc758 \ud569\uc9d1\ud569\n\\[C_N = I_1 \\cup I_2 \\cup I_3 \\cup \\cdots \\cup I_{2^N}\\]\n\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4. \\[2^N \\cdot 2\\delta < \\frac{\\epsilon}{2}\\]\uc778 \\(\\delta > 0\\)\uc744 \ud0dd\ud558\uc790. \uadf8\ub9ac\uace0 \uac01 \\(I_i = [a_i ,\\, b_i ]\\)\uc5d0 \ub300\ud558\uc5ec \uc5f4\ub9b0 \uad6c\uac04 \\(J_i = (a_i – \\delta ,\\, b_i + \\delta )\\)\ub97c \uc0dd\uac01\ud558\uc790. \uadf8\ub7ec\uba74 \\(I_i \\subseteq J_i\\)\uc774\uba70 \\(J_i\\)\ub4e4\uc758 \uae38\uc774\uc758 \ud569\uc740 [\\(I_i\\)\ub4e4\uc758 \uae38\uc774\uc758 \ud569]\uacfc [\uad6c\uac04\uc758 \uac1c\uc218\uc5d0 \\(2\\delta\\)\ub97c \uacf1\ud55c \uac12]\uc744 \ub354\ud55c \uac83\uacfc \uac19\ub2e4. \uc989 \\(J_i\\)\ub4e4\uc758 \uae38\uc774\uc758 \ud569\uc740
      \n\\[\\left( \\frac{2}{3} \\right)^N + 2^N \\cdot 2\\delta < \\frac{\\epsilon}{2} + \\frac{\\epsilon}{2} = \\epsilon \\]\n\uc774\ub2e4. \\(C \\subseteq C_N\\)\uc774\ubbc0\ub85c \\(C\\)\ub294 \\(J_i\\)\ub4e4\uc5d0 \uc758\ud558\uc5ec \ub36e\uc778\ub2e4. \uadf8\ub7f0\ub370 \\(J_i\\)\ub294 \uae38\uc774\uc758 \ud569\uc774 \\(\\epsilon\\) \ubbf8\ub9cc\uc778 \uc5f4\ub9b0 \uad6c\uac04\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(C\\)\ub294 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub2e4.\n<\/p>\n

      [2] \uad6c\uac04 \\([0,\\,1]\\)\uc5d0 \uc18d\ud55c \uc2e4\uc218\ub97c \uc0bc\uc9c4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74 \uc774 \uc218\ub4e4\uc740
      \n\\[0. x_1 x_2 x_3 \\cdots _{(3)}\\]
      \n\uc640 \uac19\uc774 \uac01 \uc790\ub9ac\uc758 \uc22b\uc790\uac00 \\(0,\\) \\(1,\\) \\(2\\) \uc911 \ud558\ub098\uc778 \uc18c\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \ub9cc\uc57d \uc774 \uc218\uac00 \\(0\\)\uc774 \uc544\ub2c8\uace0 \uc774 \ud45c\ud604\uc774 \uc720\ud55c\uc18c\uc218\ub77c\uba74 \uac00\uc7a5 \uc791\uc740 \uc790\ub9ac\uc758 \uc22b\uc790\uc5d0\uc11c \\(1\\)\uc744 \ube7c\uace0 \uadf8 \ub4a4\uc5d0 \\(2\\)\ub97c \uc5f0\ub2ec\uc544 \uc801\uc74c\uc73c\ub85c\uc368 \ubb34\ud55c\uc18c\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc608\ucee8\ub300
      \n\\[0.1022_{(3)}\\]
      \n\ub77c\uba74 \uc774 \uc18c\uc218\ub294
      \n\\[0.1021222\\cdots_{(3)}\\]
      \n\uc640 \uac19\uc740 \ubb34\ud55c\uc18c\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n

      \uc774\uc81c \uce78\ud1a0\uc5b4 \uc9d1\ud569\uc5d0 \uc18d\ud55c \uc218\ub97c \uc0dd\uac01\ud558\uc790. \\(C_1\\)\uc744 \ub9cc\ub4e4 \ub54c \uac00\uc6b4\ub370 \\(1\/3\\)\uc744 \uc81c\uac70\ud588\uc73c\ubbc0\ub85c \uc0bc\uc9c4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0c8\uc744 \ub54c \uc18c\uc218\uc810 \uc544\ub798 \uccab\uc9f8 \uc790\ub9ac\uac00 \\(1\\)\uc778 \uc218\ub294 \uc81c\uc678\ub41c\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(C_2\\)\ub97c \ub9cc\ub4e4 \ub54c, \uc774\uc804 \ub2e8\uacc4\uc5d0\uc11c \ub9cc\ub4e4\uc5b4\uc9c4 \ub450 \uad6c\uac04\uc758 \uac00\uc6b4\ub370 \\(1\/3\\)\uc744 \uc81c\uac70\ud588\uc73c\ubbc0\ub85c \uc0bc\uc9c4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0c8\uc744 \ub54c \uc18c\uc218\uc810 \uc544\ub798 \ub458\uc9f8 \uc790\ub9ac\uac00 \\(1\\)\uc778 \uc218\ub294 \uc81c\uc678\ub41c\ub2e4. \uc774\uc640 \uac19\uc740 \uacfc\uc815\uc5d0 \uc758\ud558\uc5ec \uc18c\uc218\uc810 \uc544\ub798 \uc22b\uc790 \uc911\uc5d0\uc11c \\(1\\)\uc774 \ud3ec\ud568\ub41c \uc218\ub294 \ubaa8\ub450 \uc81c\uc678\ub41c\ub2e4. \uc989 \uce78\ud1a0\uc5b4 \uc9d1\ud569\uc740 \uc0bc\uc9c4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0c8\uc744 \ub54c \uc18c\uc218\uc810 \uc544\ub798 \uc22b\uc790\uac00 \\(0\\)\uacfc \\(2\\) \ubfd0\uc778 \uc218\uc758 \ubaa8\uc784\uc774\ub2e4.<\/p>\n

      \uce78\ud1a0\uc5b4 \uc9d1\ud569\uc774 \uac00\uc0b0\uc9d1\ud569\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(C\\)\uc758 \uc6d0\uc18c\ub97c
      \n\\[ \\begin{align}
      \nx_1 &= x_{1,1} \\, x_{1,2} \\, x_{1,3} \\cdots _{(3)} \\\\[6pt]
      \nx_2 &= x_{2,1} \\, x_{2,2} \\, x_{2,3} \\cdots _{(3)} \\\\[6pt]
      \nx_3 &= x_{3,1} \\, x_{3,2} \\, x_{3,3} \\cdots _{(3)} \\\\[6pt]
      \n&\\,\\,\\vdots
      \n\\end{align}\\]
      \n\uacfc \uac19\uc774 \ud55c \uc904\ub85c \ubaa8\ub450 \ub098\uc5f4\ud560 \uc218 \uc788\ub2e4. \uc774\uc81c \ub2e4\uc74c\uacfc \uac19\uc740 \uc218\uc5f4\uc744 \uc0dd\uac01\ud558\uc790.
      \n\\[y_i = \\begin{cases}
      \n0 & \\quad \\text{if} \\,\\, x_{i,i} = 2 \\\\[6pt]
      \n2 & \\quad \\text{if} \\,\\, x_{i,i} = 0 \\\\[6pt]
      \n\\end{cases}\\]
      \n\uadf8\ub9ac\uace0
      \n\\[y = y_1 \\, y_2 \\, y_3 \\, \\cdots _{(3)}\\]
      \n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uac01 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(y_i \\ne x_{i,i}\\)\uc774\ubbc0\ub85c \uc784\uc758\uc758 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(y \\ne x_i\\)\uc774\ub2e4. \uc989 \\(y\\)\ub294 \uc5b4\ub5a0\ud55c \\(x_i\\)\uc640\ub3c4 \uac19\uc9c0 \uc54a\ub2e4. \uadf8\ub7ec\ub098 \\(y\\)\ub294 \uc0bc\uc9c4\ubc95\uc73c\ub85c \ub098\ud0c0\ub0c8\uc744 \ub54c \uc18c\uc218\uc810 \uc544\ub798 \ubaa8\ub4e0 \uc790\ub9ac\uc758 \uc22b\uc790\uac00 \\(0\\) \ub610\ub294 \\(2\\)\uc778 \uc218\uc774\ubbc0\ub85c \\(C\\)\uc758 \uc6d0\uc18c\uc774\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(C\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      <\/p>\n

      \ud568\uc218\uc758 \uc9c4\ub3d9\ub7c9<\/h3>\n

      \ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \\(x\\)\uac00 \\(c\\)\ub97c \uc9c0\ub0a0 \ub54c \ud568\uc22b\uac12 \\(f(x)\\)\ub3c4 \ubd80\ub4dc\ub7fd\uac8c \ubcc0\ud654\ud55c\ub2e4. \uadf8\ub7ec\ub098 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\uba74 \\(x\\)\uac00 \\(c\\)\ub97c \uc9c0\ub098\ub294 \uc21c\uac04 \\(f(x)\\)\uc758 \uac12\uc774 \uae09\uaca9\ud558\uac8c \ubcc0\ud654\ud55c\ub2e4. \uc774\ub54c \\(c\\)\uc5d0\uc11c \\(f(x)\\)\uc758 \uac12\uc774 \ubcc0\ud654\ud558\ub294 \uc591\uc744 \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uc9c4\ub3d9\ub7c9\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc815\ud655\ud55c \uc815\uc758\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

      \n

      \uc815\uc758 2.<\/span>
      \n\ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \ud558\uc790.<\/p>\n

        \n
      1. \\(J\\)\uac00 \uad6c\uac04\uc774\uace0 \\([a,\\,b] \\cap J \\ne \\varnothing\\)\uc77c \ub54c, \\(J\\)\uc5d0\uc11c \\(f\\)\uc758 \uc9c4\ub3d9\ub7c9<\/span>(oscillation)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
        \n\\[\\varOmega_f (J) := \\sup \\left\\{ f(x) -f(y) \\,\\vert\\, x\\in J \\cap [a,\\,b] ,\\,\\, y\\in J \\cap [a,\\,b] \\right\\}.\\]\n<\/li>\n
      2. \uc810 \\(c\\in [a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc9c4\ub3d9\ub7c9<\/span>\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
        \n\\[\\omega_f (c) := \\lim_{h \\to 0^+} \\varOmega _f ((t-h ,\\, t+h )).\\]\n<\/li>\n<\/ol>\n<\/div>\n

        \uc9c1\uad00\uc801\uc73c\ub85c \uad6c\uac04 \\(J\\)\uc5d0\uc11c \ud568\uc218\uc758 \uc9c4\ub3d9\ub7c9\uc774\ub780 \\(J\\)\uc5d0\uc11c \ud568\uc22b\uac12\uc758 \ucd5c\ub300 \ubcc0\ud654\ub7c9\uc774\uba70, \uc810 \\(c\\)\uc5d0\uc11c \ud568\uc218\uc758 \uc9c4\ub3d9\ub7c9\uc774\ub780 \\(x\\)\uac00 \\(c\\)\ub97c \uc9c0\ub098\ub294 \uc21c\uac04 \\(f(x)\\)\uc758 \ubcc0\ud654\ub7c9\uc774\ub2e4. \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uc9c4\ub3d9\ub7c9\uc740 \\(x\\to c\\)\uc77c \ub54c \\(f\\)\uc758 \uc0c1\uadf9\ud55c\uacfc \ud558\uadf9\ud55c\uc758 \ucc28\uc640 \uac19\ub2e4.<\/p>\n

        \n

        \uc608\uc81c 3.<\/span>
        \n\uc784\uc758\uc758 \uc2e4\uc218 \\(c\\)\uc5d0\uc11c \ucd5c\ub300\uc815\uc218\ud568\uc218 \\(f(x) = [x]\\)\uc758 \uc9c4\ub3d9\ub7c9\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n

        \ud480\uc774.<\/span>
        \n\\(c\\)\uac00 \uc815\uc218\uac00 \uc544\ub2d0 \ub54c\uc5d0\ub294 \\(\\omega_f (c) = 0\\)\uc774\uba70, \\(c\\)\uac00 \uc815\uc218\uc77c \ub54c\uc5d0\ub294 \\(\\omega_f (c) = 1\\)\uc774\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \n

        \uc608\uc81c 4.<\/span>
        \n\ud568\uc218 \\(g\\)\uac00
        \n\\[g(x) = \\begin{cases}
        \n\\sin \\frac{1}{x} \\quad & \\text{if} \\,\\, x \\ne 0 \\\\[6pt]
        \n0 \\quad & \\text{if} \\,\\, x = 0
        \n\\end{cases}\\]
        \n\uc73c\ub85c \uc815\uc758\ub418\uc5b4 \uc788\uc744 \ub54c, \\(0\\)\uc5d0\uc11c \\(g\\)\uc758 \uc9c4\ub3d9\ub7c9\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n

        \ud480\uc774.<\/span>
        \n\\(h > 0\\)\uc774\uace0 \\(J = (-h ,\\, h )\\)\uc77c \ub54c, \\(J\\)\uc5d0\uc11c \\(g\\)\uc758 \ucd5c\ub313\uac12\uc740 \\(1\\)\uc774\uace0 \ucd5c\uc19f\uac12\uc740 \\(-1\\)\uc774\ub2e4. \uc989 \\(\\varOmega_g (J) = 2\\)\uc774\ub2e4.<\/p>\n

        \uadf8\ub7ec\ubbc0\ub85c \\(\\omega_g (0) = 2\\)\uc774\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \uc774\uc81c \uc9c4\ub3d9\ub7c9\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc774 \uc131\uc9c8\ub4e4\uc740 \ub4a4\uc5d0\uc11c \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\ub97c \uc99d\uba85\ud558\ub294 \ub370\uc5d0 \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n

        \n

        \ubcf4\uc870\uc815\ub9ac 1.<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc774\uba74 \\([a,\\,b]\\)\uc758 \uc784\uc758\uc758 \uc810 \\(t\\)\uc5d0\uc11c \\(f\\)\uc758 \uc9c4\ub3d9\ub7c9 \\(\\omega_f (t)\\)\uac00 \uc74c\uc774 \uc544\ub2cc \uc720\ud55c\uac12\uc73c\ub85c\uc11c \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \\(t\\in [a,\\,b]\\)\ub77c\uace0 \ud558\uc790. \\(t\\in (a,\\,b)\\)\ub77c\uace0 \ud574\ub3c4 \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\ub294\ub2e4. \uad6c\uac04 \\(J\\)\uc5d0 \ub300\ud558\uc5ec
        \n\\[\\begin{align}
        \nM_J &= \\sup \\left\\{ f(x) \\,\\vert\\, x\\in J\\cap [a,\\,b] \\right\\} ,\\\\[6pt]
        \nm_J &= \\inf \\left\\{ f(x) \\,\\vert\\, x\\in J\\cap [a,\\,b] \\right\\}
        \n\\end{align}\\]
        \n\ub77c\uace0 \uc815\uc758\ud558\uc790. \\(\\sup (-f (x)) = -\\inf (f(x))\\)\uc774\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
        \n\\[\\varOmega_f (J) = M_j – m_j \\ge 0\\tag{3}\\]
        \n\\((t-h_0 ,\\, t+h_0 )\\subseteq (a,\\,b)\\)\uc778 \uc591\uc218 \\(h_0\\)\uc744 \ud0dd\ud558\uc790. \\(0 < h < h_0\\)\uc778 \\(h\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\phi (h) := \\varOmega_f ((t-h ,\\, t+h ))\\]\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\phi (h)\\)\ub294 \\((0,\\,h_0 )\\)\uc5d0\uc11c \uc99d\uac00\ud558\ubbc0\ub85c \\(0\\)\uc5d0\uc11c \uc6b0\uadf9\ud55c\uc744 \uac00\uc9c4\ub2e4. \uadf8\ub7f0\ub370 (3)\uc5d0 \uc758\ud558\uc5ec \\(\\phi(h) > 0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\omega_f (t)\\)\ub294 \uc74c\uc774 \uc544\ub2cc \uc720\ud55c\uac12\uc73c\ub85c\uc11c \uc874\uc7ac\ud55c\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \n

        \ubcf4\uc870\uc815\ub9ac 2.<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec
        \n\\[H = \\left\\{ t\\in [a,\\,b] \\,\\vert\\, \\omega_f (t) \\ge \\epsilon \\right\\}\\]
        \n\uc740 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \\(H\\)\ub294 \\([a,\\,b]\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ubbc0\ub85c \uc720\uacc4\uc774\ub2e4. \uc774\uc81c \\(H\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \ubaa8\ub4e0 \ud56d\uc774 \\(H\\)\uc5d0 \uc18d\ud558\uc9c0\ub9cc \\(H\\) \ubc16\uc758 \uc810 \\(t\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4 \\(\\left\\{ t_n \\right\\}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(\\omega_f (t) < \\epsilon\\)\uc774\ubbc0\ub85c \\(h_0 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.
        \n\\[\\varOmega_f ((t-h_0 ,\\, t+h_0 )) < \\epsilon \\tag{4}\\]\n\\(t_n \\to t\\)\uc774\ubbc0\ub85c \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec\n\\[\\left( t_N - \\frac{h_0}{2} ,\\, t_N + \\frac{h_0}{2} \\right) \\subseteq (t-h_0 ,\\, t+h_0 )\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\uba74 (4)\uc5d0 \uc758\ud558\uc5ec\n\\[\\varOmega_f \\left(\\left( t_N - \\frac{h_0}{2} ,\\, t_N + \\frac{h_0}{2} \\right) \\right) < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989 \\(\\omega_f (t_N ) < \\epsilon\\)\uc778\ub370, \uc774\uac83\uc740 \\(t_N \\in H\\)\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \\(H\\)\ub294 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4. \uc989 \\(H\\)\ub294 \uc720\uacc4\uc778 \ub2eb\ud78c \uc9d1\ud569\uc774\ubbc0\ub85c \ud558\uc774\ub124-\ubcf4\ub810 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(H\\)\ub294 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\ub2e4.\n<\/span><\/p>\n<\/div>\n

        \n

        \ubcf4\uc870\uc815\ub9ac 3.<\/span><\/p>\n

        \\(I\\)\uac00 \uc720\uacc4\uc778 \ub2eb\ud78c \uad6c\uac04\uc774\uace0 \ud568\uc218 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc720\uacc4\uc774\uba70 \\(\\epsilon > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(t\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(\\omega_f (t) < \\epsilon\\)\uc774\uba74 \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \uae38\uc774\uac00 \\(\\delta\\) \ubbf8\ub9cc\uc774\uace0 \\(I\\)\uc5d0 \ud3ec\ud568\ub418\ub294 \uc784\uc758\uc758 \ub2eb\ud78c \uad6c\uac04 \\(J\\)\uc5d0 \ub300\ud558\uc5ec \\(\\varOmega_f (J) < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \uac01 \\(t\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta_t > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec
        \n\\[\\varOmega_f ((t-\\delta_t ,\\, t+\\delta_t )) < \\epsilon \\tag{5}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(\\delta_t \/2 > 0\\)\uc774\ubbc0\ub85c \ud558\uc774\ub124-\ubcf4\ub810 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(t_1 ,\\) \\(t_2 ,\\) \\(\\cdots,\\) \\(t_N\\)\uc774 \uc874\uc7ac\ud558\uc5ec
        \n\\[I \\subseteq \\bigcup_{j=1}^N \\left( t_j – \\frac{\\delta_{t_j}}{2} ,\\, t_j + \\frac{\\delta_{t_j}}{2} \\right)\\]
        \n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4.
        \n\\[\\delta = \\frac{1}{2} \\min \\left\\{ \\delta_{t_j} \\,\\vert\\, j=1,\\,2,\\, \\cdots,\\,N \\right\\}\\]
        \n\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(J\\subseteq I\\)\uc774\uba74 \uc801\ub2f9\ud55c \\(j\\in\\left\\{ 1,\\,2,\\,\\cdots,\\,N \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec
        \n\\[J \\cap \\left( t_j – \\frac{\\delta_{t_j}}{2},\\, t_j + \\frac{\\delta_{t_j}}{2} \\right) \\ne \\varnothing\\]
        \n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub354\uc6b1\uc774, \ub9cc\uc57d \\(J\\)\uc758 \uae38\uc774\uac00 \\(\\delta\\) \ubbf8\ub9cc\uc774\uba74 \\(J \\subseteq (t_j – \\delta_{t_j} ,\\, t_j + \\delta_{t_j} )\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \ud2b9\ud788 (5)\uc5d0 \uc758\ud558\uc5ec
        \n\\[\\varOmega_f (J) \\le \\varOmega_f ((t_j – \\delta_{t_j} ,\\, t_j + \\delta_{t_j} )) < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/span><\/p>\n<\/div>\n

        \n

        \ubcf4\uc870\uc815\ub9ac 4.<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ubd88\uc5f0\uc18d\uc810\ub4e4\uc758 \ubaa8\uc784\uc744 \\(E\\)\ub77c\uace0 \ud558\uba74
        \n\\[E = \\bigcup_{j=1}^{\\infty} \\left\\{ t\\in [a,\\,b] \\,\\bigg\\vert\\, \\omega_f (t) \\ge \\frac{1}{j} \\right\\}\\]
        \n\uc774\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \\(t\\in[a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\omega_f (t) =0\\)\uc778 \uac83\uc774\ub2e4. \ub530\ub77c\uc11c \\(t\\in E\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\omega_f (t) > 0\\)\uc774\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \uc774\uc81c \ube44\ub85c\uc18c \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\ub97c \uc99d\uba85\ud560 \uc900\ube44\uac00 \ub418\uc5c8\ub2e4.<\/p>\n

        <\/p>\n

        \ub974\ubca0\uadf8\uc758 \uc815\ub9ac<\/h3>\n

        \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\ub4e4\uc774 \uc544\uc8fc \uc801\uc740 \ubd80\ubd84\ub9cc \ucc28\uc9c0\ud558\uba74 \\(f\\)\uac00 \uc774 \uad6c\uac04\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\ub294 \uc815\ub9ac\uc774\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 2. (\ub974\ubca0\uadf8\uc758 \uc815\ub9ac)<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\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 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\ub4e4\uc758 \uc9d1\ud569\uc758 \uce21\ub3c4\uac00 \\(0\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\ub4e4\uc758 \ubaa8\uc784\uc744 \\(E\\)\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \\(E\\)\uc758 \uce21\ub3c4\uac00 \\(0\\)\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \ubcf4\uc870\uc815\ub9ac 4\uc5d0 \uc758\ud558\uc5ec \\(j_0 \\in \\mathbb{N}\\)\uc774 \uc874\uc7ac\ud558\uc5ec
        \n\\[H = \\left\\{ t\\in [a,\\,b] \\,\\bigg\\vert\\, \\omega_f (t) \\ge \\frac{1}{j_0} \\right\\}\\]
        \n\uc758 \uce21\ub3c4\uac00 \\(0\\)\uc774 \uc544\ub2c8\ub2e4. \ud2b9\ud788 \\(\\epsilon _0 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(H\\)\ub97c \ub36e\uc73c\uba74\uc11c \ubaa8\ub4e0 \uc6d0\uc18c\uac00 \uad6c\uac04\uc778 \uc784\uc758\uc758 \uc9d1\ud569 \\(\\left\\{ I_k \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec
        \n\\[\\sum_{k=1}^{\\infty} \\left\\lvert I_k \\right\\rvert \\ge \\epsilon_0 \\tag{6}\\]
        \n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(P = \\left\\{ x_0 ,\\, x_1 ,\\, \\cdots ,\\, x_n \\right\\}\\)\uc774 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\((x_{k-1} ,\\, x_k )\\cap H \\ne \\varnothing\\)\uc774\uba74 \\(H\\)\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec
        \n\\[M_k (f,\\,P) – m_k (f,\\,P) \\ge \\frac{1}{j_0}\\]
        \n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
        \n\\[\\begin{align}
        \nU(f,\\,P) – L(f,\\,P)
        \n&= \\sum_{k=1}^{n} (M_k (f,\\,P) – m_k (f,\\,P)) \\Delta x_k \\\\[6pt]
        \n&\\ge \\sum_{(x_{k-1},\\,x_k )\\cap H \\ne \\varnothing} (M_k (f,\\,P) – m_k (f,\\,P)) \\Delta x_k \\\\[6pt]
        \n&\\ge \\frac{1}{j_0} \\sum_{(x_{k-1} ,\\,x_k ) \\cap H \\ne \\varnothing} \\Delta x_k
        \n\\end{align}\\]
        \n\uadf8\ub7f0\ub370
        \n\\[\\left\\{ [x_{k-1} ,\\, x_k ] \\,\\vert\\, (x_{k-1} ,\\,x_k ) \\cap H \\ne \\varnothing \\right\\}\\]
        \n\uc740 \\(H\\)\ub97c \ub36e\ub294 \uad6c\uac04\ub4e4\uc758 \ubaa8\uc784\uc774\ubbc0\ub85c (6)\uc5d0 \uc758\ud558\uc5ec
        \n\\[U(f,\\,P) – L(f,\\,P) \\ge \\frac{\\epsilon_0}{j_0} > 0\\]
        \n\uc744 \uc5bb\ub294\ub2e4. \uc774\uac83\uc740 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\ub294 \ub370\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(E\\)\ub294 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub2e4.<\/p>\n

        \ub2e4\uc74c\uc73c\ub85c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(E\\)\uac00 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\ud55c\uc744 \\(M,\\) \ud558\ud55c\uc744 \\(m\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\((M-m+b-a) < j_0 \\epsilon\\)\uc778 \uc790\uc5f0\uc218 \\(j_0\\)\uc744 \ud0dd\ud558\uc790. \\(E\\)\uac00 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ubbc0\ub85c\n\\[H = \\left\\{ t\\in [a,\\,b] \\,\\bigg\\vert\\, \\omega_f (t) \\ge \\frac{1}{j_0} \\right\\}\\]\n\ub3c4 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub2e4. \ub530\ub77c\uc11c \\(H\\)\ub97c \ub36e\uc73c\uba74\uc11c \uae38\uc774\uc758 \ud569\uc774 \\(1\/j_0\\) \ubbf8\ub9cc\uc778 \uc5f4\ub9b0 \uad6c\uac04\ub4e4\uc758 \ubaa8\uc784 \\(\\left\\{ I_\\nu \\right\\}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \ubcf4\uc870\uc815\ub9ac 2\uc5d0 \uc758\ud558\uc5ec \\(N\\)\uac1c\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c4 \ubd80\ubd84\uc9d1\ud569 \\(\\left\\{ I_1 ,\\, I_2 ,\\, \\cdots ,\\, I_N \\right\\}\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \uc9d1\ud569\uc774 \\(H\\)\ub97c \ub36e\uc73c\uba70\n\\[\\sum_{\\nu = 1}^{N} \\left\\lvert I_\\nu \\right\\rvert < \\frac{1}{j_0} \\tag{7}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774\uc81c \\(U(f,\\,P) - L(f,\\,P) < \\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubd84\ud560 \\(P\\)\ub97c \ucc3e\uc544\uc57c \ud55c\ub2e4. \\(I_\\nu\\)\ub4e4\uc758 \ub05d\uc810\ub4e4\uc758 \ubaa8\uc784\uc774 \uadf8\ub7ec\ud55c \ubd84\ud560\uc758 \uc6d0\uc18c\uac00 \ub41c\ub2e4. \uadf8\ub7ec\ub098 \uadf8\ub7ec\ud55c \uc810\ub4e4\ub9cc \ubaa8\uc73c\uba74 \ucda9\ubd84\ud558\uc9c0 \uc54a\uc73c\uba70 \\(I_\\nu\\)\uc5d0 \uc758\ud574 \ub36e\uc774\uc9c0 \uc54a\ub294 \\([a,\\,b]\\)\uc758 \ubd80\ubd84\uc744 \ubd84\ud560\ud558\uc5ec \uc810\uc744 \ub354 \ucd94\uac00\ud574\uc57c \ud55c\ub2e4. \uc989\n\\[\\tilde{I} \\subseteq [a,\\,b] \\setminus \\bigcup_\\nu I_\\nu := [a,\\,b] \\setminus \\left( I_1 \\cup I_2 \\cup \\cdots \\cup I_N \\right)\\]\n\uc774\ub77c\uace0 \ud558\uc790. \\(I_\\nu\\)\ub4e4\uc774 \\(H\\)\ub97c \ub36e\uc73c\ubbc0\ub85c \uc784\uc758\uc758 \\(t\\in \\tilde{I}\\)\uc5d0 \ub300\ud558\uc5ec \\(\\omega_f (t) < 1\/j_0\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \ubcf4\uc870\uc815\ub9ac 3\uc5d0 \uc758\ud558\uc5ec \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(J\\subseteq \\tilde{I} ,\\) \\(\\lvert J \\rvert < \\delta\\)\uc77c \ub54c\ub9c8\ub2e4 \\(\\varOmega_f (J) < 1\/j_0 \\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

        \\([a,\\,b] \\setminus \\bigcup_\\nu I_\\nu \\)\ub97c \uae38\uc774\uac00 \\(\\delta\\) \ubbf8\ub9cc\uc778 \uad6c\uac04\ub4e4 \\(J_1 ,\\) \\(J_2 ,\\) \\(\\cdots ,\\) \\(J_s \\)\ub85c \ubd84\ud560\ud558\uc790. \uadf8\ub7ec\uba74 \uac01 \\(p\\)\uc5d0 \ub300\ud558\uc5ec
        \n\\[\\varOmega_f (J_p ) < \\frac{1}{j_0} \\tag{8}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(I_\\nu\\)\uc758 \ub05d\uc810\ub4e4\uacfc \\(I_p\\)\uc758 \ub05d\uc810\ub4e4\uc744 \ubaa8\uc544\uc11c \ub9cc\ub4e0 \ubd84\ud560\uc744\n\\[P = \\left\\{ x_0 ,\\, x_1 ,\\, \\cdots ,\\, x_n \\right\\}\\]\n\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\((x_{k-1} ,\\, x_k ) \\cap H \\ne \\varnothing\\)\uc774\uba74 \\(x_{k-1}\\)\uacfc \\(x_k\\)\ub294 \uc801\ub2f9\ud55c \\(I_\\nu\\)\uc758 \ub05d\uc810\uc774\ubbc0\ub85c (7)\uc5d0 \uc758\ud558\uc5ec\n\\[\\sum_{(x_{k-1} ,\\,x_k ) \\cap H \\ne\\varnothing} (M_k (f,\\,P) - m_k (f,\\,P)) \\Delta x_k \\le \\frac{M-m}{j_0} \\tag{9}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub9cc\uc57d \\((x_{k-1} ,\\,x_k ) \\cap H = \\varnothing\\)\uc774\uba74 \\(x_{k-1}\\)\uacfc \\(x_k\\)\ub294 \uc801\ub2f9\ud55c \\(J_p\\)\uc758 \ub05d\uc810\uc774\ubbc0\ub85c (8)\uc5d0 \uc758\ud558\uc5ec\n\\[\\sum_{(x_{k-1} ,\\,x_k ) \\cap H = \\varnothing} (M_k ( f,\\,P) - m_k ( f ,\\,P)) \\Delta x_k \\le \\frac{1}{j_0} \\sum_{k=1}^n \\Delta x_k = \\frac{b-a}{j_0} \\tag{10}\\]\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub85c\uc368 (9)\uc640 (10)\uc744 \uacb0\ud569\ud558\uba74\n\\[\\begin{align}\nU(f,\\,P) - L(f,\\,P)\n&= \\sum_{k=1}^{n} (M_k (f,\\,P) - m_k (f,\\,P)) \\Delta x_k \\\\[6pt]\n&\\le \\frac{M-m+b-a}{j_0} < \\epsilon\n\\end{align}\\]\n\uc774\ubbc0\ub85c \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.\n<\/span><\/p>\n<\/div>\n

        <\/p>\n

        \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\ub97c \ud65c\uc6a9\ud55c \uc608\uc81c\ub4e4<\/h3>\n

        \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74 \ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \uc27d\uac8c \ud310\ubcc4\ud560 \uc218 \uc788\ub2e4.<\/p>\n

        \n

        \uc608\uc81c 5.<\/span>
        \n\\(D = \\left\\{ 1\/k \\,\\vert\\, k \\in \\mathbb{N} \\right\\}\\)\uc774\ub77c\uace0 \ud558\uace0, \ud568\uc218 \\(f : [0,\\,1] \\to \\mathbb{R}\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \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\uc2dc\uc624.\n<\/p>\n

        \ud480\uc774.<\/span>
        \n\\([0,\\,1]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uc9d1\ud569\uc740
        \n\\[E = \\left\\{ 0 \\right\\} \\cup D\\]
        \n\uc774\ub2e4. \\(E\\)\ub294 \uac00\uc0b0\uc9d1\ud569\uc774\ubbc0\ub85c \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \n

        \uc608\uc81c 6.<\/span>
        \n\ud568\uc218 \\(g\\)\uac00
        \n\\[g(x) = \\begin{cases}
        \n1 \\quad & \\text{if} \\,\\, x\\in \\mathbb{Q} \\\\[6pt]
        \n0 \\quad & \\text{if} \\,\\, x\\notin \\mathbb{Q}
        \n\\end{cases}\\]
        \n\ub85c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(g\\)\uac00 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \ubd88\uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.\n<\/p>\n

        \ud480\uc774.<\/span>
        \n\\(g\\)\ub294 \\([0,\\,1]\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\ub2e4. \uadf8\ub7f0\ub370 \uc608\uc81c 1\uc5d0 \uc758\ud558\uc5ec \\([0,\\,1]\\)\uc740 \uce21\ub3c4\uac00 \\(0\\)\uc774 \uc544\ub2c8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(g\\)\ub294 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \n

        \uc608\uc81c 7.<\/span>
        \n\\(C\\)\uac00 \uce78\ud1a0\uc5b4 \uc9d1\ud569\uc774\uace0 \ud568\uc218 \\(h : [0,\\,1] \\to \\mathbb{R}\\)\uac00
        \n\\[h(x) = \\begin{cases}
        \n1 \\quad & \\text{if} \\,\\, x\\in C \\\\[6pt]
        \n0 \\quad & \\text{if} \\,\\, x\\notin C
        \n\\end{cases}\\]
        \n\ub85c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\([0,\\,1]\\)\uc5d0\uc11c \\(h\\)\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud558\uc2dc\uc624.\n<\/p>\n

        \ud480\uc774.<\/span>
        \n\\(h\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc740 \ubaa8\ub450 \\(C\\)\uc5d0 \uc18d\ud55c\ub2e4. \uadf8\ub7f0\ub370 \uc608\uc81c 2\uc5d0 \uc758\ud558\uc5ec \\(C\\)\ub294 \uce21\ub3c4\uac00 \\(0\\)\uc778 \uc9d1\ud569\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(h\\)\ub294 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        \n

        \uc608\uc81c 8.<\/span>
        \n\ud568\uc218 \\(f : [a,\\,b] \\to [c,\\,d]\\)\uac00 \uc77c\ub300\uc77c \ub300\uc751\uc774\uace0 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \uc5ed\ud568\uc218 \\(f^{-1}\\)\ub294 \\([c,\\,d]\\)\uc5d0\uc11c \uc801\ubd84 \ubd88\uac00\ub2a5\ud55c \ud568\uc218 \\(f\\)\uc758 \uc608\ub97c \ub4dc\uc2dc\uc624.\n<\/p>\n

        \ud480\uc774.<\/span>
        \n\\([a,\\,b] = [0,\\,1]\\)\uc774\uace0 \\(C\\)\uac00 \uce78\ud1a0\uc5b4 \uc9d1\ud569(Cantor set<\/a>)\uc774\uba70 \\(D\\)\uac00 SVC-\uc9d1\ud569(Smith-Volterra-Cantor set<\/a>)\uc774\ub77c\uace0 \ud558\uc790. \uc989 \\(C\\)\ub294 \uc608\uc81c 2\uc5d0\uc11c \uc815\uc758\ud55c \uce78\ud1a0\uc5b4 \uc9d1\ud569\uc774\uba70, \\(D\\)\ub294 \uad6c\uac04 \\([0,\\,1]\\)\uc5d0\uc11c \uc2dc\uc791\ud558\uc5ec \ub0a8\uc740 \uad6c\uac04\uc5d0\uc11c \uac00\uc6b4\ub370 \\(1\/4\\)\uc529\uc744 \uc81c\uac70\ud558\uc5ec \uc5bb\uc740 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4.<\/p>\n

        \uba3c\uc800 \ud568\uc218 \\(f : C \\to D\\)\ub97c \uc21c\uc99d\uac00\ud558\ub294 \uc77c\ub300\uc77c \ub300\uc751\uc774 \ub418\ub3c4\ub85d \uc815\uc758\ud55c\ub2e4. \uc989 \\(C\\)\ub97c \ub9cc\ub4dc\ub294 \\(n\\)\ubc88\uc9f8 \ub2e8\uacc4\uc640 \\(D\\)\ub97c \ub9cc\ub4dc\ub294 \\(n\\)\ubc88\uc9f8 \ub2e8\uacc4\uc5d0\uc11c \uac01\uac01 \ub2eb\ud78c \uad6c\uac04\ub4e4\uc758 \ud569\uc9d1\ud569 \\(C_n\\)\uacfc \\(D_n\\)\uc744 \uc5bb\ub294\ub370, \uad6c\uac04\uc758 \ub05d\uc810\uc758 \uc218\uac00 \uac19\uc73c\ubbc0\ub85c \uc774 \uc810\ub4e4\uc774 \uc11c\ub85c \ub300\uc751\ub418\ub3c4\ub85d \uc21c\uc99d\uac00\ud568\uc218\ub97c \uc815\uc758\ud560 \uc218 \uc788\uc73c\uba70, \uac01 \uc870\uac01\ub4e4(\ub2eb\ud78c \uad6c\uac04\ub4e4)\uc758 \ub0b4\ubd80\uc5d0\uc11c \uc77c\ucc28\ud568\uc218\uac00 \ub418\ub3c4\ub85d \ud568\uc73c\ub85c\uc368 \\(C\\)\ub85c\ubd80\ud130 \\(D\\)\ub85c\uc758 \uc21c\uc99d\uac00\ud558\ub294 \uc77c\ub300\uc77c \ub300\uc751 \\(f\\)\ub97c \uad6c\uc131\ud560 \uc218 \uc788\ub2e4.<\/p>\n

        \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\([0,\\,1] \\setminus C_n\\)\uacfc \\([0,\\,1] \\setminus D_n\\)\ub3c4 \uac01\uac01 \uac1c\uc218\uac00 \uac19\uc740 \uc5f4\ub9b0 \uad6c\uac04\ub4e4\uc758 \ud569\uc9d1\ud569\uc774\ubbc0\ub85c \uc5f4\ub9b0 \uad6c\uac04\ub4e4\uc758 \ub05d\uc810\uc744 \ub300\uc751\uc2dc\ud0a4\ub418 \uc21c\uc11c\uac00 \ubc18\ub300\uac00 \ub418\ub3c4\ub85d \ud558\uba70, \uac01 \uc5f4\ub9b0 \uad6c\uac04\uc758 \ub0b4\ubd80\uc5d0\uc11c \uac10\uc18c\ud558\ub294 \uc77c\ucc28\ud568\uc218\uac00 \ub418\ub3c4\ub85d \ud568\uc73c\ub85c\uc368 \ub450 \uc9d1\ud569 \uc0ac\uc774\uc5d0 \uc21c\uac10\uc18c\ud558\ub294 \uc77c\ub300\uc77c \ub300\uc751\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\([0,\\,1] \\setminus C\\)\ub97c \\([0,\\,1] \\setminus D\\)\uc5d0 \ub300\uc751\uc2dc\ud0a8 \ud568\uc218\ub97c \\(f\\)\ub77c \ud558\uc790.<\/p>\n

        \uc774\uc81c \\(f\\)\ub294 \\([0,\\,1]\\)\ub85c\ubd80\ud130 \\([0,\\,1]\\)\ub85c\uc758 \uc77c\ub300\uc77c \ub300\uc751\uc774\uba70, \\(C\\)\uc758 \uc810\uc744 \\(D\\)\uc5d0 \ub300\uc751\uc2dc\ud0a8\ub2e4. \ub610\ud55c \\(f\\)\ub294 \\(C\\) \uc704\uc5d0\uc11c\ub9cc \ubd88\uc5f0\uc18d\uc774\uace0 \\(C\\)\uac00 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774\ubbc0\ub85c \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \uadf8\ub7ec\ub098 \\(f^{-1}\\)\ub294 \\(D\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\uace0 \\(D\\)\uc758 \uce21\ub3c4\uac00 \\(0\\)\uc774 \uc544\ub2c8\ubbc0\ub85c(\ucc38\uc870: Wikipedia<\/a>) \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(f^{-1}\\)\ub294 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4.
        \n<\/span><\/p>\n<\/div>\n

        <\/p>\n

        <\/h3>\n

        <\/p>\n

        <\/p>\n

        <\/p>\n

        <\/p>\n","protected":false},"excerpt":{"rendered":"

        \uc774 \ud3ec\uc2a4\ud2b8\ub294 \ubbf8\uc801\ubd84\ud559\ubcf4\ub2e4 \uc0c1\uae09 \uacfc\uc815\uc758 \ub0b4\uc6a9\uc744 \ub2e4\ub8e8\uace0 \uc788\uc2b5\ub2c8\ub2e4. \ubbf8\uc801\ubd84\ud559\uc744 \ucc98\uc74c \uacf5\ubd80\ud558\ub294 \ud559\uc0dd\ub4e4\uc740 \uc774 \ud3ec\uc2a4\ud2b8\uc758 \ub0b4\uc6a9\uc744 \uc774\ud574\ud558\uae30 \uc5b4\ub824\uc6b8 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc758 \ub0b4\uc6a9\uc744 \uc774\ud574\ud558\uae30 \uc704\ud574\uc11c\ub294 \ub9ac\ub9cc \uc801\ubd84\uc758 \uc5c4\ubc00\ud55c \uc815\uc758, \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\uc131\uc5d0 \ub300\ud55c \ub9ac\ub9cc \ud310\uc815\ubc95, \uc0c1\ud55c\uacfc \ud558\ud55c\uc758 \uc131\uc9c8\uc744 \uc54c\uc544\uc57c \ud569\ub2c8\ub2e4. \ubbf8\uc801\ubd84\ud559\uc744 \ucc98\uc74c \uacf5\ubd80\ud558\uc9c0\ub9cc \uc774 \ud3ec\uc2a4\ud2b8\uc758 \ub0b4\uc6a9\uc744 \uaf2d \uc54c\uace0 \uc2f6\uc740 \uc0ac\ub78c\uc740 \uc815\uc758 1, \uc815\ub9ac 1, \uc608\uc81c 1, \uc815\ub9ac 2\uc758 \ub0b4\uc6a9(\ud480\uc774\uc640 \uc99d\uba85 \uc81c\uc678)\uacfc \uc608\uc81c 5, \uc608\uc81c 6\uc744…<\/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":[318,312,315,314,142,290,311,289,316,313,317],"class_list":["post-3665","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","category-mathematical-analysis","tag-cantor-set","tag-lebesgues-theorem","tag-measure-0","tag-measure-zero","tag-oscillation","tag-riemann-integral","tag-311","tag-289","tag-316","tag-313","tag-317"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3665","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=3665"}],"version-history":[{"count":35,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3665\/revisions"}],"predecessor-version":[{"id":8327,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3665\/revisions\/8327"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=3665"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=3665"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=3665"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}