{"id":9484,"date":"2025-10-20T18:53:59","date_gmt":"2025-10-20T09:53:59","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9484"},"modified":"2026-09-27T14:38:23","modified_gmt":"2026-09-27T05:38:23","slug":"ch04-limit-of-functions-and-continuity","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch04-limit-of-functions-and-continuity\/","title":{"rendered":"\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131"},"content":{"rendered":"<div class=\"analysis2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131<\/h2>\n\n --><\/p>\n<p>\uc55e \uc7a5\uc5d0\uc11c\ub294 \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \uc218\uc5f4\uc758 \uc218\ub834, \uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569, \ucef4\ud329\ud2b8\uc131\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \uac1c\ub150\uc744 \ubc14\ud0d5\uc73c\ub85c \uac70\ub9ac\uacf5\uac04 \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131\uc744 \ub2e4\ub8ec\ub2e4.<\/p>\n<h3>\ud568\uc218\uc758 \uadf9\ud55c<\/h3>\n<p>\uac70\ub9ac\uacf5\uac04 \\((X,d_X)\\), \\((Y,d_Y)\\)\uc640 \ud568\uc218 \\(f\\colon X\\to Y\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc810 \\(a\\in X\\)\uac00 \\(X\\)\uc758 \uc9d1\uc801\uc810\uc774\uace0 \\(L\\in Y\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud574 \\(\\delta&gt;0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(0&lt;d_X(x,a)&lt;\\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(d_Y(f(x),L)&lt;\\varepsilon\\)\uc774 \uc131\ub9bd\ud558\uba74, \u201c\\(x\\to a\\)\uc77c \ub54c \\(f(x)\\)\uac00 \\(L\\)\ub85c \uc218\ub834\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\uac83\uc744 \uae30\ud638\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[<br \/>\n\\lim_{x\\to a}f(x)=L<br \/>\n\\quad\\text{\ub610\ub294}\\quad<br \/>\nf(x)\\to L\\quad(x\\to a).<br \/>\n\\]<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 4.1. (\ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc720\uc77c\uc131)<\/span><\/p>\n<p>\\(a\\)\uac00 \\(X\\)\uc758 \uc9d1\uc801\uc810\uc77c \ub54c \\(\\displaystyle\\lim_{x\\to a}f(x)\\)\uac00 \uc874\uc7ac\ud558\uba74 \uadf8 \uac12\uc740 \uc720\uc77c\ud558\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uc11c\ub85c \ub2e4\ub978 \ub450 \uc810 \\(L,M\\in Y\\)\uac00 \ubaa8\ub450 \uadf9\ud55c\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \\(\\varepsilon=d_Y(L,M)\/3\\)\uc73c\ub85c \ub450\uba74 \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[<br \/>\n0&lt;d_X(x,a)&lt;\\delta<br \/>\n\\]<br \/>\n\uc774\uba74 \ub3d9\uc2dc\uc5d0 \\(d_Y(f(x),L)&lt;\\varepsilon\\), \\(d_Y(f(x),M)&lt;\\varepsilon\\)\uc774\ub2e4. \\(a\\)\uac00 \uc9d1\uc801\uc810\uc774\ubbc0\ub85c \uc774\ub7ec\ud55c \\(x\\)\ub97c \ud558\ub098 \ud0dd\ud560 \uc218 \uc788\uace0, \uc0bc\uac01\ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\nd_Y(L,M)\\le d_Y(L,f(x))+d_Y(f(x),M)&lt;2\\varepsilon,<br \/>\n\\]<br \/>\n\uc778\ub370 \uc774\ub294 \\(\\varepsilon=d_Y(L,M)\/3\\)\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc218\uc5f4\uc744 \uc0ac\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uadf9\ud55c\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ud2b9\uc131\ud654\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 4.2. (\ud558\uc774\ub124 \uc815\ub9ac)<\/span><\/p>\n<p>\\(X\\)\uc640 \\(Y\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(f\\colon X\\to Y\\)\uac00 \ud568\uc218\uc774\uba70 \\(a\\)\uac00 \\(X\\)\uc758 \uc9d1\uc801\uc810\uc774\uace0 \\(L\\in Y\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc740 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\displaystyle\\lim_{x\\to a}f(x)=L\\)<\/li>\n<li>\\(x_n\\to a\\), \\(x_n\\neq a\\), \\(x_n\\in X\\)\uc778 \ubaa8\ub4e0 \uc218\uc5f4 \\(\\{x_n\\}\\)\uc5d0 \ub300\ud574 \\(f(x_n)\\to L\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n(1)\uc744 \uac00\uc815\ud558\uace0 \\(x_n\\to a\\), \\(x_n\\neq a\\)\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uadf9\ud55c\uc758 \uc815\uc758\uc5d0\uc11c \uc5bb\ub294 \\(\\delta&gt;0\\)\ub97c \ud0dd\ud55c\ub2e4. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(d_X(x_n,a)&lt;\\delta\\)\uc774\uace0 \\(x_n\\neq a\\)\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\nd_Y(f(x_n),L)&lt;\\varepsilon.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(f(x_n)\\to L\\)\uc774\ub2e4.<\/p>\n<p>\uc5ed\uc73c\ub85c (2)\ub97c \uac00\uc815\ud558\uace0 (1)\uc774 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc5b4\ub5a4 \\(\\varepsilon_0&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 \\(n\\)\ub9c8\ub2e4<br \/>\n\\[<br \/>\n0&lt;d_X(x_n,a)&lt;\\frac1n,<br \/>\n\\quad<br \/>\nd_Y(f(x_n),L)\\ge\\varepsilon_0<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(x_n\\in X\\)\ub97c \ud0dd\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\uba74 \\(x_n\\to a\\)\uc774\uace0 \\(x_n\\neq a\\)\uc774\uc9c0\ub9cc \\(f(x_n)\\not\\to L\\)\uc774\ubbc0\ub85c (2)\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ud558\uc774\ub124 \uc815\ub9ac\ub294 \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc74c\uc744 \ubcf4\uc77c \ub54c \uc720\uc6a9\ud558\ub2e4. \uc989 \\(x\\to a\\)\uc77c \ub54c \\(f(x)\\)\uc758 \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc74c\uc744 \ubcf4\uc774\uae30 \uc704\ud574, \\(s_n\\to a\\), \\(t_n\\to a\\)\uc774\uc9c0\ub9cc \\(\\{f(s_n)\\}\\)\uacfc \\(\\{f(t_n)\\}\\)\uc774 \ub2e4\ub978 \uac12\uc5d0 \uc218\ub834\ud558\ub294 \ub450 \uc218\uc5f4 \\(\\{s_n\\}\\), \\(\\{t_n\\}\\)\uc744 \ucc3e\uc73c\uba74 \ub41c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.3.<\/span><\/p>\n<p>\\(X\\)\uac00 \uc9d1\ud569\uc774\uace0 \\(A\\subseteq X\\)\ub77c\uace0 \ud558\uc790. \\(A\\)\uc758 <span class=\"defined\">\ud2b9\uc131\ud568\uc218<\/span>(characteristic function) \\(\\chi_A\\colon X\\to\\{0,1\\}\\)\ub294 \\(x\\in A\\)\uc77c \ub54c \\(\\chi_A(x)=1\\)\uc774\uace0, \\(x\\notin A\\)\uc77c \ub54c \\(\\chi_A(x)=0\\)\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218\uc774\ub2e4.<\/p>\n<p>\\(X=\\mathbb R\\), \\(A=\\mathbb Q\\)\uc77c \ub54c \\(\\chi_{\\mathbb Q}\\)\ub294 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(a\\in\\mathbb R\\)\uc77c \ub54c \\(a\\)\ub85c \uc218\ub834\ud558\uace0 \\(s_n\\neq a\\)\uc778 \uc720\ub9ac\uc218\uc5f4 \\(\\{s_n\\}\\)\uacfc \\(a\\)\ub85c \uc218\ub834\ud558\uace0 \\(t_n\\neq a\\)\uc778 \ubb34\ub9ac\uc218\uc5f4 \\(\\{t_n\\}\\)\uc744 \ud0dd\ud558\uba74 \\(\\chi_{\\mathbb Q}(s_n)\\to1\\)\uc774\uc9c0\ub9cc \\(\\chi_{\\mathbb Q}(t_n)\\to0\\)\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<\/div>\n<p>\uc2e4\ud568\uc218\uc758 \uacbd\uc6b0 <span class=\"defined\">\ud55c\ubc29\ud5a5 \uadf9\ud55c<\/span>(one-sided limit)\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \\(X\\subseteq\\mathbb R\\), \\(f\\colon X\\to\\mathbb R\\), \\(a\\in\\mathbb R\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<ul>\n<li>\\(a\\)\uac00 \\(X\\cap(-\\infty,a)\\)\uc758 \uc9d1\uc801\uc810\uc77c \ub54c, \\(f\\)\ub97c \\(X\\cap(-\\infty,a)\\)\ub85c \uc81c\ud55c\ud55c \ud568\uc218\uc758 \\(a\\)\uc5d0\uc11c\uc758 \uadf9\ud55c\uc744 \\(f\\)\uc758 <span class=\"defined\">\uc88c\uadf9\ud55c<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(\\displaystyle\\lim_{x\\to a-}f(x)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(x\\in X\\), \\(0&lt;a-x&lt;\\delta\\)\uc774\uba74 \\(|f(x)-L|&lt;\\varepsilon\\)\uc77c \ub54c \\(\\displaystyle\\lim_{x\\to a-}f(x)=L\\)\uc774\ub77c\uace0 \ud55c\ub2e4.<\/li>\n<li>\\(a\\)\uac00 \\(X\\cap(a,\\infty)\\)\uc758 \uc9d1\uc801\uc810\uc77c \ub54c, \uac19\uc740 \ubc29\ubc95\uc73c\ub85c <span class=\"defined\">\uc6b0\uadf9\ud55c<\/span> \\(\\displaystyle\\lim_{x\\to a+}f(x)\\)\ub97c \uc815\uc758\ud55c\ub2e4. \uc989 \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(x\\in X\\), \\(0&lt;x-a&lt;\\delta\\)\uc774\uba74 \\(|f(x)-L|&lt;\\varepsilon\\)\uc77c \ub54c \\(\\displaystyle\\lim_{x\\to a+}f(x)=L\\)\uc774\ub77c\uace0 \ud55c\ub2e4.<\/li>\n<\/ul>\n<p>\ud2b9\ud788 \\(a\\)\uac00 \\(X\\cap(-\\infty,a)\\)\uc640 \\(X\\cap(a,\\infty)\\)\uc758 \uc9d1\uc801\uc810\uc77c \ub54c, \\(\\displaystyle\\lim_{x\\to a}f(x)=L\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc774 \ubaa8\ub450 \uc874\uc7ac\ud558\uace0 \ub458 \ub2e4 \\(L\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.1.<\/span><br \/>\n\\(D\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(a\\in D&#8217;\\), \\(f\\colon D\\to\\mathbb R\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[<br \/>\n\\lim_{x\\to a}f(x)=+\\infty<br \/>\n\\]<br \/>\n\ub77c\ub294 \uac83\uc740 \uc784\uc758\uc758 \\(M&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(x\\in D\\), \\(0&lt;d_X(x,a)&lt;\\delta\\)\uc774\uba74 \\(f(x)&gt;M\\)\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4. \\(\\displaystyle\\lim_{x\\to a}f(x)=-\\infty\\)\ub3c4 \uac19\uc740 \ubc29\uc2dd\uc73c\ub85c \uc815\uc758\ud558\uc2dc\uc624. \ub610\ud55c \uac01\uac01\uc758 \uc815\uc758\uac00 \ud558\uc774\ub124 \uc815\ub9ac\uc640 \uac19\uc740 \uc218\uc5f4\uc5d0 \uc758\ud55c \ud2b9\uc131\ud654\ub97c \uac00\uc9d0\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 4.2.<\/span><br \/>\n\\(D\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(c\\in D&#8217;\\)\uc774\uba70 \ud568\uc218 \\(f\\colon D\\to\\mathbb R\\)\uc640 \\(g\\colon D\\to\\mathbb R\\)\uac00 \\(x\\to c\\)\uc77c \ub54c \uac01\uac01 \\(A\\), \\(B\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(k\\)\uac00 \uc2e4\uc218\uc778 \uc0c1\uc218\uc774\uba74, \\(x\\to c\\)\uc77c \ub54c \\((kf)(x)\\to kA\\)\uc774\ub2e4.<\/li>\n<li>\\(x\\to c\\)\uc77c \ub54c \\((f+g)(x)\\to A+B\\)\uc774\ub2e4.<\/li>\n<li>\\(x\\to c\\)\uc77c \ub54c \\((fg)(x)\\to AB\\)\uc774\ub2e4.<\/li>\n<li>\\(B\\ne0\\)\uc774\uba74, \\(c\\)\uc758 \ucda9\ubd84\ud788 \uc791\uc740 \uad6c\uba4d\ub6ab\ub9b0 \uadfc\ubc29\uc5d0\uc11c\ub294 \\(g(x)\\ne0\\)\uc774\uace0 \uadf8\uacf3\uc5d0\uc11c \uc815\uc758\ub41c \ubaab\uc5d0 \ub300\ud558\uc5ec \\(x\\to c\\)\uc77c \ub54c \\((f\/g)(x)\\to A\/B\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.3.<\/span><br \/>\n\\(D\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(a\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\uba70 \ud568\uc218 \\(f\\colon D\\to\\mathbb R^d\\)\uc758 \\(j\\)\ubc88\uc9f8 \uc88c\ud45c\ub97c \ub098\ud0c0\ub0b4\ub294 \ud568\uc218\uac00 \\(f_j\\)\ub77c\uace0 \ud558\uc790. \uc989<br \/>\n\\[<br \/>\nf(x)=(f_1(x),f_2(x),\\ldots,f_d(x))<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(L=(L_1,L_2,\\ldots,L_d)\\in\\mathbb R^d\\)\ub77c\uace0 \ud558\uc790. \\(x\\to a\\)\uc77c \ub54c \\(f(x)\\to L\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(j=1,2,\\ldots,d\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\to a\\)\uc77c \ub54c \\(f_j(x)\\to L_j\\)\uc778 \uac83\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 4.4.<\/span><br \/>\n\\((x,y)\\to(0,0)\\)\uc77c \ub54c, \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\uac00 \uc218\ub834\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f(x,y)=\\displaystyle\\frac{5xy^2}{x^2+y^2}\\)<\/li>\n<li>\\(f(x,y)=\\displaystyle\\frac{3xy}{x^2+y^2}\\)<\/li>\n<li>\\(f(x,y)=\\displaystyle\\frac{x^2y}{x^2+y^2}\\)<\/li>\n<\/ol>\n<\/div>\n<h3>\uc5f0\uc18d\ud568\uc218<\/h3>\n<p>\ud568\uc218 \\(f\\colon X\\to Y\\)\uac00 \uc810 \\(a\\in X\\)\uc5d0\uc11c <span class=\"defined\">\uc5f0\uc18d<\/span>(continuous)\uc774\ub77c\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(d_X(x,a)&lt;\\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(d_Y(f(x),f(a))&lt;\\varepsilon\\)\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4. \uc815\uc758\uc5ed\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218\ub97c <span class=\"defined\">\uc5f0\uc18d\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(E\\)\uac00 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(a\\in E\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(B'(a,r)\\cap E=\\varnothing\\)\uc778 \uc591\uc218 \\(r\\)\uc774 \uc874\uc7ac\ud558\uba74, \\(a\\)\ub97c \\(E\\)\uc758 <span class=\"defined\">\uace0\ub9bd\uc810<\/span>(isolated point)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 4.4. (\uc218\uc5f4\uc744 \uc0ac\uc6a9\ud55c \uc5f0\uc18d\uc758 \uc815\uc758)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\colon X\\to Y\\)\uc640 \uc810 \\(a\\in X\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc740 \ubaa8\ub450 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f\\)\uac00 \\(a\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\\(x_n\\to a\\), \\(x_n\\in X\\)\uc778 \ubaa8\ub4e0 \uc218\uc5f4 \\(\\{x_n\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x_n)\\to f(a)\\)\uc774\ub2e4.<\/li>\n<li>\\(a\\)\uac00 \\(X\\)\uc758 \uace0\ub9bd\uc810\uc774\uac70\ub098, \\(\\displaystyle\\lim_{x\\to a}f(x)=f(a)\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n(1)\uc744 \uac00\uc815\ud558\uace0 \\(x_n\\to a\\)\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uc5f0\uc18d\uc131\uc5d0\uc11c \uc5bb\ub294 \\(\\delta&gt;0\\)\ub97c \ud0dd\ud558\uba74, \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(d_X(x_n,a)&lt;\\delta\\)\uc774\ubbc0\ub85c \\(d_Y(f(x_n),f(a))&lt;\\varepsilon\\)\uc774\ub2e4. \ub530\ub77c\uc11c (2)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\ubc18\ub300\ub85c (2)\ub97c \uac00\uc815\ud558\uace0 \\(f\\)\uac00 \\(a\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc5b4\ub5a4 \\(\\varepsilon_0&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 \\(n\\)\ub9c8\ub2e4<br \/>\n\\[<br \/>\nd_X(x_n,a)&lt;\\frac1n,<br \/>\n\\quad<br \/>\nd_Y(f(x_n),f(a))\\ge\\varepsilon_0<br \/>\n\\]<br \/>\n\uc778 \\(x_n\\in X\\)\ub97c \ud0dd\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\uba74 \\(x_n\\to a\\)\uc774\uc9c0\ub9cc \\(f(x_n)\\not\\to f(a)\\)\uc774\ubbc0\ub85c \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c (1)\uacfc (2)\ub294 \ub3d9\uce58\uc774\ub2e4.<\/p>\n<p>\ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(a\\)\uac00 \uace0\ub9bd\uc810\uc774\uba74 \ucda9\ubd84\ud788 \uc791\uc740 \\(\\delta&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(B(a,\\delta)=\\{a\\}\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \uc790\ub3d9\uc73c\ub85c \\(a\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \\(a\\)\uac00 \uc9d1\uc801\uc810\uc774\uba74 \uc5f0\uc18d\uc131\uc758 \uc815\uc758\uc640 \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0\uc11c \ucc28\uc774\ub294 \\(x=a\\)\ub97c \ud5c8\uc6a9\ud558\ub294\uc9c0\ubfd0\uc774\uace0, \\(x=a\\)\uc5d0\uc11c\ub294 \\(d_Y(f(x),f(a))=0\\)\uc774\ub2e4. \ub530\ub77c\uc11c (1)\uacfc (3)\ub3c4 \ub3d9\uce58\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.5.<\/span><br \/>\n\\(I\\)\uac00 \uad6c\uac04\uc774\uace0 \ud568\uc218 \\(f\\colon I\\to\\mathbb R\\)\uac00 \ub2e8\uc870\ud568\uc218\ub77c\uace0 \ud558\uc790. \\(I\\)\uc758 \ub0b4\ubd80\uc810\uc5d0\uc11c\ub294 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc774 \ubaa8\ub450 \uc874\uc7ac\ud558\uace0, \ub05d\uc810\uc5d0\uc11c\ub294 \uc815\uc758\uc5ed \uc548\ucabd\uc5d0\uc11c\uc758 \ud55c\ubc29\ud5a5 \uadf9\ud55c\uc774 \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \ub610\ud55c \\(I\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\uac00 \ub9ce\uc544\uc57c \uac00\uc0b0\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \ub2e8\uc870\uc99d\uac00\uc778 \uacbd\uc6b0 \uac01 \ubd88\uc5f0\uc18d\uc810 \\(c\\)\uc5d0 \ub300\ud558\uc5ec \\((f(c-),f(c+))\\) \uc548\uc758 \uc720\ub9ac\uc218\ub97c \ud558\ub098 \ud0dd\ud558\ub294 \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/p>\n<\/div>\n<p>\uc790\uc8fc \ub9cc\ub098\ub294 \uc5f0\uc18d\ud568\uc218\uc640 \uc5f0\uc18d\ud568\uc218\ub97c \ub9cc\ub4dc\ub294 \uae30\ubcf8 \ubc29\ubc95\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4. \uc544\ub798 \uc131\uc9c8 \uc911 \uc77c\ubd80\ub294 \uc774\uc5b4\uc9c0\ub294 \ubb38\uc81c\uc5d0\uc11c \uc9c1\uc811 \uc99d\uba85\ud55c\ub2e4.<\/p>\n<ul>\n<li>\ud56d\ub4f1\ud568\uc218 \\(\\operatorname{id}_X\\colon X\\to X\\)\uc640 \uc0c1\uc218\ud568\uc218\ub294 \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\\(f\\colon X\\to Y\\)\uc640 \\(g\\colon Y\\to Z\\)\uac00 \uc5f0\uc18d\uc774\uba74 \\(g\\circ f\\colon X\\to Z\\)\ub3c4 \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\\(X_1\\times X_2\\)\uc5d0 <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch02-metric-spaces\">2\uc7a5<\/a>\uc5d0\uc11c \uc815\uc758\ud55c \uacf1\uac70\ub9ac \uc911 \ud558\ub098\ub97c \uc8fc\uba74 \uc0ac\uc601\ud568\uc218 \\(\\pi_i\\colon X_1\\times X_2\\to X_i\\)\ub294 \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\ub178\ub984\uacf5\uac04 \\((V,\\|\\cdot\\|)\\)\uc5d0\uc11c \\(x\\mapsto\\|x\\|\\)\ub294 \uc5f0\uc18d\uc774\ub2e4. \uc2e4\uc81c\ub85c \uc5ed\uc0bc\uac01\ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud574 \\(\\bigl|\\|x\\|-\\|y\\|\\bigr|\\le\\|x-y\\|\\)\uc774\ub2e4.<\/li>\n<li>\uac70\ub9ac\uacf5\uac04 \\((X,d)\\)\uc5d0\uc11c \uac70\ub9ac\ud568\uc218 \\(d\\colon X\\times X\\to\\mathbb R\\)\ub294 \uacf1\uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\\(f\\)\uc640 \\(g\\)\uac00 \uc5f0\uc18d\uc778 \uc2e4\ud568\uc218\uc774\uba74 \\(f+g\\), \\(f-g\\), \\(fg\\)\ub294 \uc5f0\uc18d\uc774\ub2e4. \ub610\ud55c \\(g(a)\\ne0\\)\uc778 \uc810 \\(a\\)\uc5d0\uc11c\ub294 \\(f\/g\\)\ub3c4 \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<\/ul>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.6.<\/span><br \/>\n\\(X\\), \\(Y\\), \\(Z\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(f\\colon X\\to Y\\)\uc640 \\(g\\colon Y\\to Z\\)\uac00 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(a\\in X&#8217;\\)\uc774\uba70, \\(x\\to a\\)\uc77c \ub54c \\(f(x)\\to b\\in Y\\)\uc774\uace0 \\(g\\)\uac00 \\(b\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<br \/>\n\\[<br \/>\n\\lim_{x\\to a}g(f(x))<br \/>\n=g\\left(\\lim_{x\\to a}f(x)\\right).<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 4.7.<\/span><br \/>\n\ubc11\uc774 \uc790\uc5f0\uc0c1\uc218 \\(e\\)\uc778 \ub85c\uadf8\ub97c <span class=\"defined\">\uc790\uc5f0\ub85c\uadf8<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \\(\\ln\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \\(x&gt;0\\)\uc77c \ub54c \\(\\ln x=\\log_e x\\)\uc774\ub2e4. \uc774 \ubb38\uc81c\uc5d0\uc11c\ub294 \ubbf8\uc801\ubd84\ud559\uc5d0\uc11c \uc775\ud78c \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \uae30\ubcf8 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4. \ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\displaystyle\\lim_{x\\to\\infty}\\left(1+\\frac1x\\right)^x\\)<\/li>\n<li>\\(\\displaystyle\\lim_{x\\to0}(1+x)^{1\/x}\\)<\/li>\n<li>\\(\\displaystyle\\lim_{x\\to0}\\frac{\\ln(1+x)}{x}\\)<\/li>\n<li>\\(\\displaystyle\\lim_{x\\to0}\\frac{e^x-1}{x}\\)<\/li>\n<li>\\(\\displaystyle\\lim_{x\\to0}\\frac{\\log_a(1+x)}{x}\\) (\ub2e8, \\(a&gt;0\\), \\(a\\ne1\\).)<\/li>\n<li>\\(\\displaystyle\\lim_{x\\to0}\\frac{a^x-1}{x}\\) (\ub2e8, \\(a&gt;0\\), \\(a\\ne1\\).)<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 4.8.<\/span><br \/>\n\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\displaystyle\\lim_{x\\to0}\\frac{\\sin x}{x}\\)<\/li>\n<li>\\(\\displaystyle\\lim_{x\\to0}\\frac{1-\\cos x}{x}\\)<\/li>\n<\/ol>\n<\/div>\n<h3>\uc5f0\uc18d\ud568\uc218\uc758 \uc704\uc0c1\uc801 \ud2b9\uc131<\/h3>\n<p>\uc5f4\ub9b0\uc9d1\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc5f0\uc18d\uc131\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 4.5. (\uc5f4\ub9b0\uc9d1\ud569\uc744 \uc0ac\uc6a9\ud55c \uc5f0\uc18d\uc758 \uc815\uc758)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\colon X\\to Y\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(Y\\)\uc758 \ubaa8\ub4e0 \uc5f4\ub9b0\uc9d1\ud569 \\(V\\)\uc5d0 \ub300\ud574 \\(f^{-1}(V)\\)\uac00 \\(X\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(f\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(V\\)\uac00 \\(Y\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(x\\in f^{-1}(V)\\)\uc774\uba74 \\(f(x)\\in V\\)\uc774\uace0, \uc5b4\ub5a4 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(B_Y(f(x),\\varepsilon)\\subseteq V\\)\uc774\ub2e4. \\(f\\)\uac00 \\(x\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\ny\\in B_X(x,\\delta)<br \/>\n\\quad\\Longrightarrow\\quad<br \/>\nf(y)\\in B_Y(f(x),\\varepsilon)\\subseteq V.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(B_X(x,\\delta)\\subseteq f^{-1}(V)\\)\uc774\uace0 \\(f^{-1}(V)\\)\ub294 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4.<\/p>\n<p>\uc5ed\uc73c\ub85c \\(Y\\)\uc758 \ubaa8\ub4e0 \uc5f4\ub9b0\uc9d1\ud569 \\(V\\)\uc5d0 \ub300\ud558\uc5ec \\(f^{-1}(V)\\)\uac00 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(a\\in X\\), \\(\\varepsilon&gt;0\\)\ub97c \ud0dd\ud558\uba74 \\(V=B_Y(f(a),\\varepsilon)\\)\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc774\uace0 \\(a\\in f^{-1}(V)\\)\uc774\ub2e4. \ub530\ub77c\uc11c \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(B_X(a,\\delta)\\subseteq f^{-1}(V)\\)\uc774\ub2e4. \uc989 \\(d_X(x,a)&lt;\\delta\\)\uc774\uba74 \\(d_Y(f(x),f(a))&lt;\\varepsilon\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(a\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \\(a\\)\uac00 \uc784\uc758\uc774\ubbc0\ub85c \\(f\\)\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc704 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud558\uba74 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 4.6. (\ub2eb\ud78c\uc9d1\ud569\uacfc \uc5f0\uc18d\ud568\uc218\uc758 \uad00\uacc4)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\colon X\\to Y\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(F\\)\uac00 \\(Y\\)\uc758 \ub2eb\ud78c\uc9d1\ud569\uc774\uba74 \\(f^{-1}(F)\\)\ub294 \\(X\\)\uc758 \ub2eb\ud78c\uc9d1\ud569\uc774\ub2e4.<\/li>\n<li>\\(K\\)\uac00 \\(X\\)\uc758 \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569\uc774\uba74 \\(f(K)\\)\ub294 \\(Y\\)\uc758 \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.9.<\/span><br \/>\n\ub2eb\ud78c\uc9d1\ud569\uacfc \uc5f0\uc18d\ud568\uc218\uc758 \uad00\uacc4(\uc815\ub9ac 4.6)\ub97c \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.10.<\/span><br \/>\n\\(X\\), \\(Y\\), \\(Z\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \ud568\uc218 \\(f\\colon X\\to Y\\)\uc640 \\(g\\colon Y\\to Z\\)\uac00 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ud569\uc131\ud568\uc218 \\(g\\circ f\\colon X\\to Z\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 4.11.<\/span><br \/>\n\ud568\uc218 \\(g\\colon\\mathbb R\\times\\mathbb R\\to\\mathbb R\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\uc744 \ub54c, \\(g\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(g(x,y)=x+y\\)<\/li>\n<li>\\(g(x,y)=xy\\)<\/li>\n<li>\\(g(x,y)=|x-y|\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.12.<\/span><br \/>\n\\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(f,g\\colon X\\to\\mathbb R\\)\uac00 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(x\\mapsto(f(x),g(x))\\)\uac00 \\(X\\)\ub85c\ubd80\ud130 \\(\\mathbb R^2\\)\ub85c\uc758 \uc5f0\uc18d\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \ub610\ud55c \ubb38\uc81c 4.10\uacfc \ubb38\uc81c 4.11\uc758 \uacb0\uacfc\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(f+g\\)\uc640 \\(fg\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.13.<\/span><br \/>\n\\(X\\)\uac00 \uac70\ub9ac\ud568\uc218 \\(d\\)\ub97c \uac00\uc9c4 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(X\\times X\\)\uac00 \uc720\ud074\ub9ac\ub4dc \uacf1\uac70\ub9ac \uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(g(x,y)=d(x,y)\\)\ub77c\uace0 \uc815\uc758\ub41c \ud568\uc218 \\(g\\colon X\\times X\\to\\mathbb R\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \ub9cc\uc57d \\(X\\times X\\)\uc5d0 \uc720\ud074\ub9ac\ub4dc \uacf1\uac70\ub9ac\uac00 \uc544\ub2cc \ub2e4\ub978 \uacf1\uac70\ub9ac\uac00 \uc8fc\uc5b4\uc838 \uc788\ub2e4\uba74 \uacb0\uacfc\uac00 \uc5b4\ub5bb\uac8c \ub2ec\ub77c\uc9c0\ub294\uc9c0 \ub17c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.14.<\/span><br \/>\n\\(X\\)\uac00 \ucef4\ud329\ud2b8 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(Y\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uba70 \\(f\\colon X\\to Y\\)\uac00 \uc77c\ub300\uc77c\ub300\uc751\uc778 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \uc5ed\ud568\uc218 \\(f^{-1}\\colon Y\\to X\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uacf5\uc5ed\uc774 \\(\\mathbb R\\)\uc778 \uc5f0\uc18d\ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub294 \ub2e4\uc74c \uc815\ub9ac\uac00 \uc720\uc6a9\ud558\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 4.7. (\ucd5c\ub300 \ucd5c\uc18c \uc815\ub9ac)<\/span><\/p>\n<p>\\(f\\colon X\\to\\mathbb R\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(K\\)\uac00 \\(X\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f\\)\ub294 \\(K\\)\uc5d0\uc11c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uc815\ub9ac 4.6\uc5d0 \uc758\ud574 \\(f(K)\\)\ub294 \\(\\mathbb R\\)\uc758 \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569\uc774\ub2e4. \ub530\ub77c\uc11c \\(f(K)\\)\ub294 \ub2eb\ud600 \uc788\uace0 \uc720\uacc4\uc774\ub2e4. \\(K\\neq\\varnothing\\)\uc774\ubbc0\ub85c \\(f(K)\\neq\\varnothing\\)\uc774\uace0, \uc644\ube44\uc131\uc5d0 \uc758\ud574 \\(\\alpha=\\sup f(K)\\), \\(\\beta=\\inf f(K)\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc0c1\ud55c\uacfc \ud558\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud574 \\(f(K)\\)\uc758 \uc810\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc774 \uac01\uac01 \\(\\alpha\\), \\(\\beta\\)\ub85c \uc218\ub834\ud558\uba70, \\(f(K)\\)\uac00 \ub2eb\ud600 \uc788\uc73c\ubbc0\ub85c \\(\\alpha,\\beta\\in f(K)\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(f\\)\ub294 \\(K\\)\uc5d0\uc11c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<h3>\uc5f0\uacb0\uc131<\/h3>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uac00 <span class=\"defined\">\uc5f0\uacb0<\/span>(connected)\ub418\uc5b4 \uc788\ub2e4\ub294 \uac83\uc740 \\(X=U\\cup V\\), \\(U\\cap V=\\varnothing\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc11c\ub85c\uc18c\uc778 \ub450 \ube44\uc5b4 \uc788\uc9c0 \uc54a\uc740 \uc5f4\ub9b0\uc9d1\ud569 \\(U,V\\subseteq X\\)\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\ub294 \ub73b\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.15.<\/span><br \/>\n\uac70\ub9ac\uacf5\uac04 \\(X\\)\uac00 \uc5f0\uacb0\ub41c \uacf5\uac04\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(X\\)\uc5d0\uc11c \uc5f4\ub824 \uc788\uc73c\uba74\uc11c \ub3d9\uc2dc\uc5d0 \ub2eb\ud600 \uc788\ub294 \uc9d1\ud569\uc774 \\(\\varnothing\\)\uacfc \\(X\\)\ubfd0\uc778 \uac83\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 4.8. (\\(\\mathbb R\\)\uc5d0\uc11c \uc5f0\uacb0\ub41c \uc9d1\ud569\uc758 \ud615\ud0dc)<\/span><\/p>\n<p>\\(\\mathbb R\\)\uc758 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \ubd80\ubd84\uc9d1\ud569\uc774 \uc5f0\uacb0\ub418\uc5b4 \uc788\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uad6c\uac04\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.16.<\/span><br \/>\n\uc815\ub9ac 4.8\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc5f0\uacb0\uc131\uc740 \uc5f0\uc18d\ud568\uc218\uc5d0 \uc758\ud574 \ubcf4\uc874\ub41c\ub2e4. \uc989, \\(f\\colon X\\to Y\\)\uac00 \uc5f0\uc18d\uc774\uace0 \\(X\\)\uac00 \uc5f0\uacb0\ub418\uc5b4 \uc788\uc73c\uba74 \\(f(X)\\)\ub3c4 \uc5f0\uacb0\ub418\uc5b4 \uc788\ub2e4. \uc774\ub85c\ubd80\ud130 \uc0ac\uc787\uac12 \uc815\ub9ac\uac00 \ub530\ub77c\uc628\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 4.9. (\uc0ac\uc787\uac12 \uc815\ub9ac)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\colon X\\to\\mathbb R\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(X\\)\uac00 \uc5f0\uacb0\ub41c \uacf5\uac04\uc774\uba74 \\(f(X)\\)\ub294 \uad6c\uac04\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.17.<\/span><br \/>\n\ud568\uc218 \\(f\\colon X\\to Y\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(X\\)\uac00 \uc5f0\uacb0\ub41c \uacf5\uac04\uc77c \ub54c \\(f(X)\\)\ub3c4 \uc5f0\uacb0\ub41c \uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \ub610\ud55c \uc774 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\ub9ac 4.9\ub97c \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uac00 <span class=\"defined\">\uacbd\ub85c\uc5f0\uacb0<\/span>(path-connected)\ub418\uc5b4 \uc788\ub2e4\ub294 \uac83\uc740 \uc784\uc758\uc758 \ub450 \uc810 \\(x,y\\in X\\)\uc5d0 \ub300\ud574 \uc5f0\uc18d\ud568\uc218 \\(\\gamma\\colon[0,1]\\to X\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\gamma(0)=x\\), \\(\\gamma(1)=y\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.18.<\/span><br \/>\n\uac70\ub9ac\uacf5\uac04 \\(X\\)\uac00 \uacbd\ub85c\uc5f0\uacb0\ub41c \uacf5\uac04\uc774\uba74 \\(X\\)\ub294 \uc5f0\uacb0\ub41c \uacf5\uac04\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \uc815\ub9ac 4.8\uacfc \uc5f0\uc18d\ud568\uc218\uac00 \uc5f0\uacb0\uc131\uc744 \ubcf4\uc874\ud55c\ub2e4\ub294 \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.19.<\/span><br \/>\n\\(\\mathbb R^2\\)\uc758 \ub450 \ubd80\ubd84\uc9d1\ud569 \\(A\\), \\(B\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<br \/>\n\\[<br \/>\nA=\\{(0,y)\\mid y\\in\\mathbb R\\},<br \/>\n\\quad<br \/>\nB=\\left\\{(x,y)\\;\\middle|\\;y=\\sin\\frac1x,\\;x&gt;0\\right\\}.<br \/>\n\\]<br \/>\n\uc774\ub54c \\(A\\cup B\\)\ub294 \uc5f0\uacb0\ub41c \uc9d1\ud569\uc774\uc9c0\ub9cc \uacbd\ub85c\uc5f0\uacb0\ub41c \uc9d1\ud569\uc740 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc810 \\(x\\in X\\)\uac00 \uc18d\ud558\ub294 \ubaa8\ub4e0 \uc5f0\uacb0\ub41c \ubd80\ubd84\uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uc744 \\(x\\)\uc758 <span class=\"defined\">\uc5f0\uacb0\uc131\ubd84<\/span>(connected component)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ud569\uc9d1\ud569\uc774 \uc11c\ub85c\uc18c\uc778 \ub450 \ube44\uc5b4 \uc788\uc9c0 \uc54a\uc740 \uc5f4\ub9b0 \ubd80\ubd84\uc9d1\ud569\uc73c\ub85c \ubd84\ub9ac\ub41c\ub2e4\uace0 \uac00\uc815\ud558\uba74, \\(x\\)\ub97c \ud3ec\ud568\ud558\ub294 \ubaa8\ub4e0 \uc5f0\uacb0\ub41c \ubd80\ubd84\uc9d1\ud569\uc740 \\(x\\)\uac00 \uc18d\ud55c \ud55c\ucabd\uc5d0\ub9cc \ub193\uc5ec\uc57c \ud558\ubbc0\ub85c \ub2e4\ub978 \ucabd\uc740 \ube44\uac8c \ub41c\ub2e4. \ub530\ub77c\uc11c \uc5f0\uacb0\uc131\ubd84 \uc790\uccb4\ub3c4 \uc5f0\uacb0\ub418\uc5b4 \uc788\ub2e4. \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(x\\)\uac00 \uc18d\ud558\ub294 \ubaa8\ub4e0 \uacbd\ub85c\uc5f0\uacb0\ub41c \ubd80\ubd84\uc9d1\ud569\uc758 \ud569\uc9d1\ud569\uc744 \\(x\\)\uc758 <span class=\"defined\">\uacbd\ub85c\uc5f0\uacb0\uc131\ubd84<\/span>(path component)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub450 \uc810\uc744 \uac01\uac01 \\(x\\)\uc640 \uc787\ub294 \uacbd\ub85c\ub97c \uc774\uc5b4 \ubd99\uc774\uba74 \uc774 \ud569\uc9d1\ud569\ub3c4 \uacbd\ub85c\uc5f0\uacb0\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.20.<\/span><br \/>\n\\(f\\colon[0,1]\\to[0,1]\\)\uc774 \uc5f0\uc18d\uc774\uba74 \uace0\uc815\uc810\uc744 \uac00\uc9d0\uc744 \ubcf4\uc774\uc2dc\uc624. \uc989 \\(f(p)=p\\)\uc778 \uc810 \\(p\\)\uac00 \\([0,1]\\)\uc5d0 \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.21.<\/span><br \/>\n\ud568\uc218 \\(f\\colon(0,1]\\to\\mathbb R\\)\ub97c \\(f(x)=\\sin\\frac1x\\)\ub85c \uc815\uc758\ud560 \ub54c, \uc774 \ud568\uc218\uac00 \\([0,1]\\)\ub85c \uc5f0\uc18d\uc801\uc73c\ub85c \ud655\uc7a5\ub420 \uc218 \uc5c6\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.22.<\/span><br \/>\n\\((X,d)\\)\uac00 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\uace0 \ud568\uc218 \\(f\\colon X\\to X\\)\uac00 \uc5b4\ub5a4 \\(0\\le k&lt;1\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nd(f(x),f(y))\\le k\\,d(x,y)<br \/>\n\\]<br \/>\n\ub97c \ubaa8\ub4e0 \\(x,y\\in X\\)\uc5d0 \ub300\ud558\uc5ec \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \\(f\\)\uac00 \uc720\uc77c\ud55c \uace0\uc815\uc810\uc744 \uac00\uc9d0\uc744 \ubcf4\uc774\uc2dc\uc624. \uc989 \\(f(p)=p\\)\uc778 \uc810 \\(p\\in X\\)\uac00 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774\uac83\uc744 <span class=\"defined\">\ubc14\ub098\ud750 \uace0\uc815\uc810 \uc815\ub9ac<\/span>(Banach fixed point theorem)\ub77c\uace0 \ubd80\ub978\ub2e4. \ud55c \uc810 \\(x_0\\in X\\)\uc5d0\uc11c \uc2dc\uc791\ud558\uc5ec \\(x_{n+1}=f(x_n)\\)\uc73c\ub85c \uc815\uc758\ud55c \uc218\uc5f4\uc744 \uc0dd\uac01\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.23.<\/span><br \/>\n\\(I=[a,b]\\)\uac00 \ub2eb\ud78c\uad6c\uac04\uc774\uace0 \ud568\uc218 \\(f\\colon I\\to\\mathbb R\\)\uac00 \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \\(f\\)\uc758 \uadf8\ub798\ud504<br \/>\n\\[<br \/>\nG_f=\\{(x,f(x))\\mid x\\in I\\}<br \/>\n\\]<br \/>\n\uac00 \\(\\mathbb R^2\\)\uc5d0\uc11c \ucef4\ud329\ud2b8\uc774\uace0 \uc5f0\uacb0\ub41c \uc9d1\ud569\uc774\uba70, \ub530\ub77c\uc11c \ub2eb\ud78c\uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \ub610\ud55c \uadf8\ub798\ud504\uac00 \ub2eb\ud600 \uc788\ub2e4\ub294 \uc870\uac74\ub9cc\uc73c\ub85c\ub294 \uc5f0\uc18d\uc131\uc744 \ubcf4\uc7a5\ud560 \uc218 \uc5c6\uc74c\uc744 \ubc18\ub840\ub85c \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<h3>\uade0\ub4f1\uc5f0\uc18d\uc131<\/h3>\n<p>\ud568\uc218 \\(f\\colon X\\to Y\\)\uac00 \\(X\\)\uc5d0\uc11c <span class=\"defined\">\uade0\ub4f1\uc5f0\uc18d<\/span>(uniformly continuous)\uc774\ub77c\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud574 \\(\\delta&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \\(d_X(x,y)&lt;\\delta\\)\uc778 \ubaa8\ub4e0 \\(x,y\\in X\\)\uc5d0 \ub300\ud574 \\(d_Y(f(x),f(y))&lt;\\varepsilon\\)\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<p>\uade0\ub4f1\uc5f0\uc18d\ud568\uc218\ub294 \uc5f0\uc18d\ud568\uc218\uc774\uc9c0\ub9cc, \uadf8 \uc5ed\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<ul>\n<li>\\(f(x)=x^2\\)\uc740 \\(\\mathbb R\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uc9c0\ub9cc \uade0\ub4f1\uc5f0\uc18d\uc774 \uc544\ub2c8\ub2e4.<\/li>\n<li>\\(f(x)=\\sin x\\)\ub294 \\(\\mathbb R\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\\(f(x)=1\/x\\)\uc740 \\((0,1]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uc9c0\ub9cc \uade0\ub4f1\uc5f0\uc18d\uc774 \uc544\ub2c8\ub2e4.<\/li>\n<li>\\(f(x)=1\/x\\)\uc740 \\([2,5]\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4.<\/li>\n<\/ul>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 4.10. (\uade0\ub4f1\uc5f0\uc18d\uc5d0 \ub300\ud55c \uce78\ud1a0\uc5b4\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\\(X\\)\uc640 \\(Y\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(f\\colon X\\to Y\\)\uac00 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(X\\)\uac00 \ucef4\ud329\ud2b8 \uacf5\uac04\uc774\uba74 \\(f\\)\ub294 \\(X\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(f\\)\uac00 \\(X\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(\\varepsilon_0&gt;0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \ubaa8\ub4e0 \\(\\delta=1\/n\\)\uc5d0 \ub300\ud558\uc5ec \\(d_X(x_n,y_n)&lt;1\/n\\)\uc774\uc9c0\ub9cc \\(d_Y(f(x_n),f(y_n))\\ge\\varepsilon_0\\)\uc778 \uc810 \\(x_n,y_n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch03-limit-of-sequences\">\uc815\ub9ac 3.11\uc758 \uc810\uc5f4\ucef4\ud329\ud2b8\uc131<\/a>\uc5d0 \uc758\ud574 \\(X\\)\uac00 \ucef4\ud329\ud2b8\uc774\uba74 \\(\\{x_n\\}\\)\uc758 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4 \\(\\{x_{n_k}\\}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(x_{n_k}\\to a\\)\ub77c\uace0 \ud558\uc790. \\(d_X(x_{n_k},y_{n_k})\\to0\\)\uc774\ubbc0\ub85c \\(y_{n_k}\\to a\\)\uc774\ub2e4. \\(f\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c \\(f(x_{n_k})\\to f(a)\\)\uc774\uace0 \\(f(y_{n_k})\\to f(a)\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(d_Y(f(x_{n_k}),f(y_{n_k}))\\to0\\)\uc778\ub370, \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uade0\ub4f1\uc5f0\uc18d\ud568\uc218\uc758 \uae30\ubcf8 \uc131\uc9c8\uc744 \uc815\ub9ac\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 4.11. (\uade0\ub4f1\uc5f0\uc18d\ud568\uc218\uc758 \uae30\ubcf8 \uc131\uc9c8)<\/span><\/p>\n<p>\ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f\\colon X\\to Y\\)\uac00 \uade0\ub4f1\uc5f0\uc18d\uc774\uace0 \\(\\{x_n\\}\\)\uc774 \\(X\\)\uc758 \ucf54\uc2dc \uc218\uc5f4\uc774\uba74 \\(\\{f(x_n)\\}\\)\uc740 \\(Y\\)\uc758 \ucf54\uc2dc \uc218\uc5f4\uc774\ub2e4.<\/li>\n<li>\\(E\\)\uac00 \\(X\\)\uc5d0\uc11c \uc870\ubc00\ud558\uace0 \\(Y\\)\uac00 \uc644\ube44\uac70\ub9ac\uacf5\uac04\uc774\uba70 \\(f\\colon E\\to Y\\)\uac00 \uade0\ub4f1\uc5f0\uc18d\uc774\uba74, \\(f\\)\ub294 \uade0\ub4f1\uc5f0\uc18d\ud568\uc218 \\(\\overline f\\colon X\\to Y\\)\ub85c \uc720\uc77c\ud558\uac8c \ud655\uc7a5\ub41c\ub2e4.<\/li>\n<li>\\(f\\colon X\\to Y\\)\uc640 \\(g\\colon Y\\to Z\\)\uac00 \uade0\ub4f1\uc5f0\uc18d\uc774\uba74 \\(g\\circ f\\colon X\\to Z\\)\ub3c4 \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n(1) \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uade0\ub4f1\uc5f0\uc18d\uc131\uc5d0\uc11c \uc5bb\ub294 \\(\\delta&gt;0\\)\ub97c \ud0dd\ud55c\ub2e4. \\(\\{x_n\\}\\)\uc774 \ucf54\uc2dc \uc218\uc5f4\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \\(m,n\\)\uc5d0 \ub300\ud558\uc5ec \\(d_X(x_m,x_n)&lt;\\delta\\)\uc774\uace0, \ub530\ub77c\uc11c \\(d_Y(f(x_m),f(x_n))&lt;\\varepsilon\\)\uc774\ub2e4.<\/p>\n<p>(2) \\(x\\in X\\)\ub97c \ud0dd\ud558\uc790. \\(E\\)\uac00 \uc870\ubc00\ud558\ubbc0\ub85c \\(x_n\\in E\\), \\(x_n\\to x\\)\uc778 \uc218\uc5f4\uc744 \ud0dd\ud560 \uc218 \uc788\ub2e4. \\(\\{x_n\\}\\)\uc740 \ucf54\uc2dc \uc218\uc5f4\uc774\ubbc0\ub85c (1)\uc5d0 \uc758\ud574 \\(\\{f(x_n)\\}\\)\uc740 \\(Y\\)\uc758 \ucf54\uc2dc \uc218\uc5f4\uc774\uace0, \\(Y\\)\uac00 \uc644\ube44\uc774\ubbc0\ub85c \uc218\ub834\ud55c\ub2e4. \uc774 \uadf9\ud55c\uc744 \\(\\overline f(x)\\)\ub85c \uc815\uc758\ud55c\ub2e4. \uac19\uc740 \uc810 \\(x\\)\ub85c \uc218\ub834\ud558\ub294 \ub450 \uc218\uc5f4\uc744 \ubc88\uac08\uc544 \ubc30\uc5f4\ud558\uba74 \uc5ed\uc2dc \\(x\\)\ub85c \uc218\ub834\ud558\ubbc0\ub85c \uade0\ub4f1\uc5f0\uc18d\uc131\uc744 \uc774\uc6a9\ud558\uc5ec \ub450 \ud568\uc22b\uac12 \uc218\uc5f4\uc758 \uadf9\ud55c\uc774 \uac19\uc74c\uc744 \uc54c \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \\(\\overline f\\)\ub294 \uc798 \uc815\uc758\ub41c\ub2e4. \\(x\\in E\\)\uc5d0\uc11c\ub294 \uc0c1\uc218\uc218\uc5f4\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\uc73c\ubbc0\ub85c \\(\\overline f(x)=f(x)\\)\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\)\uc758 \uade0\ub4f1\uc5f0\uc18d\uc131\uc5d0\uc11c \\(d_X(u,v)&lt;\\delta\\)\uc774\uba74 \\(d_Y(f(u),f(v))&lt;\\varepsilon\/2\\)\uac00 \ub418\ub3c4\ub85d \\(\\delta&gt;0\\)\ub97c \ud0dd\ud55c\ub2e4. \\(d_X(x,y)&lt;\\delta\/3\\)\uc774\uba74 \\(E\\)\uc758 \uc810 \\(u_n\\to x\\), \\(v_n\\to y\\)\ub97c \ud0dd\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c \\(d_X(u_n,v_n)&lt;\\delta\\)\uac00 \ub418\uac8c \ud560 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \\(d_Y(f(u_n),f(v_n))&lt;\\varepsilon\/2\\)\uc774\uace0 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \\(d_Y(\\overline f(x),\\overline f(y))\\le\\varepsilon\/2&lt;\\varepsilon\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\overline f\\)\ub294 \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \ub450 \uc5f0\uc18d \ud655\uc7a5\uc774 \\(E\\)\uc5d0\uc11c \uc77c\uce58\ud55c\ub2e4\uba74 \uc870\ubc00\uc131\uc5d0 \uc758\ud574 \ubaa8\ub4e0 \\(X\\)\uc5d0\uc11c \uc77c\uce58\ud558\ubbc0\ub85c \uc720\uc77c\uc131\ub3c4 \ub530\ub978\ub2e4.<\/p>\n<p>(3) \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uba3c\uc800 \\(g\\)\uc758 \uade0\ub4f1\uc5f0\uc18d\uc131\uc5d0\uc11c \\(d_Y(u,v)&lt;\\eta\\)\uc774\uba74 \\(d_Z(g(u),g(v))&lt;\\varepsilon\\)\uc774 \ub418\ub294 \\(\\eta&gt;0\\)\ub97c \ud0dd\ud558\uace0, \ub2e4\uc74c\uc73c\ub85c \\(f\\)\uc758 \uade0\ub4f1\uc5f0\uc18d\uc131\uc5d0\uc11c \\(d_X(x,y)&lt;\\delta\\)\uc774\uba74 \\(d_Y(f(x),f(y))&lt;\\eta\\)\uac00 \ub418\ub294 \\(\\delta&gt;0\\)\ub97c \ud0dd\ud558\uba74 \ub41c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.24.<\/span><br \/>\n\ud568\uc218 \\(f(x)=x^2\\)\uc774 \\(\\mathbb R\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774 \uc544\ub2d8\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.25.<\/span><br \/>\n\\(X\\), \\(Y\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \uac01 \\(f_n\\colon X\\to Y\\)\uac00 \uade0\ub4f1\uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(\\{f_n\\}\\)\uc774 \\(f\\colon X\\to Y\\)\ub85c <span class=\"defined\">\uade0\ub4f1\uc218\ub834<\/span>\ud55c\ub2e4\ub294 \uac83\uc740 \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n\\ge N\\)\uc774\uace0 \\(x\\in X\\)\uc774\uba74<br \/>\n\\[<br \/>\nd_Y(f_n(x),f(x))&lt;\\varepsilon<br \/>\n\\]<br \/>\n\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4\uace0 \ud558\uc790. \\(f_n\\to f\\)\uac00 \uade0\ub4f1\uc218\ub834\uc774\uba74 \\(f\\)\ub3c4 \uade0\ub4f1\uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<h3>\uc5ec\ub7ec \uac00\uc9c0 \uc5f0\uc18d\uc131<\/h3>\n<p>\uade0\ub4f1\uc5f0\uc18d \uc678\uc5d0\ub3c4 \ud574\uc11d\ud559\uc5d0\uc11c \uc790\uc8fc \ub4f1\uc7a5\ud558\ub294 \uc5f0\uc18d\uc758 \uac1c\ub150\uc774 \uba87 \uac00\uc9c0 \uc788\ub2e4.<\/p>\n<h4>\ub9bd\uc2dc\uce20 \uc5f0\uc18d<\/h4>\n<p>\ud568\uc218 \\(f\\colon X\\to Y\\)\uac00 <span class=\"defined\">\ub9bd\uc2dc\uce20 \uc5f0\uc18d<\/span>(Lipschitz continuous)\uc774\ub77c\ub294 \uac83\uc740, \uc5b4\ub5a4 \uc0c1\uc218 \\(L\\ge0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x,y\\in X\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nd_Y(f(x),f(y))\\le L\\,d_X(x,y)<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac00\uc7a5 \uc791\uc740 \\(L\\)\uc758 \uac12\uc744 \\(f\\)\uc758 <span class=\"defined\">\ub9bd\uc2dc\uce20 \uc0c1\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\ud788 \\(L&lt;1\\)\uc778 \uacbd\uc6b0 \\(f\\)\ub97c <span class=\"defined\">\ucd95\uc18c\uc0ac\uc0c1<\/span>(contraction mapping)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<h4>\\(\\alpha\\)-\ud694\ub354 \uc5f0\uc18d<\/h4>\n<p>\\(0&lt;\\alpha\\le1\\)\uc744 \uace0\uc815\ud558\uc790. \ud568\uc218 \\(f\\colon X\\to Y\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \uc0c1\uc218 \\(C\\ge0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x,y\\in X\\)\uc5d0 \ub300\ud574<br \/>\n\\[<br \/>\nd_Y(f(x),f(y))\\le C\\,d_X(x,y)^\\alpha<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uba74 \\(f\\)\uac00 <span class=\"defined\">\\(\\alpha\\)-\ud694\ub354 \uc5f0\uc18d<\/span>(\\(\\alpha\\)-H\u00f6lder continuous)\uc774\ub77c\uace0 \ud55c\ub2e4. \ud2b9\ud788 \\(\\alpha=1\\)\uc778 \uacbd\uc6b0\uac00 \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc774\ub2e4. \ub9bd\uc2dc\uce20 \uc5f0\uc18d\ud568\uc218\uc640 \ud694\ub354 \uc5f0\uc18d\ud568\uc218\ub294 \ubaa8\ub450 \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4.<\/p>\n<p>\uba87 \uac00\uc9c0 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<ul>\n<li>\\(f(x)=\\sqrt x\\)\ub294 \\([0,1]\\)\uc5d0\uc11c \\(1\/2\\)-\ud694\ub354 \uc5f0\uc18d\uc774\ubbc0\ub85c \uade0\ub4f1\uc5f0\uc18d\uc774\uc9c0\ub9cc \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc740 \uc544\ub2c8\ub2e4.<\/li>\n<li>\\(f(x)=x^2\\)\uc740 \uc784\uc758\uc758 \\(M&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\([-M,M]\\)\uc5d0\uc11c\ub294 \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc774\uc9c0\ub9cc \\(\\mathbb R\\) \uc804\uccb4\uc5d0\uc11c\ub294 \ub9bd\uc2dc\uce20 \uc5f0\uc18d\ub3c4 \uade0\ub4f1\uc5f0\uc18d\ub3c4 \uc544\ub2c8\ub2e4.<\/li>\n<\/ul>\n<h4>\uc808\ub300\uc5f0\uc18d<\/h4>\n<p>\uc801\ubd84 \uc774\ub860\uacfc \uae4a\uc774 \uad00\ub828 \uc788\ub294 \uc5f0\uc18d\uc131 \uac1c\ub150\uc73c\ub85c <span class=\"defined\">\uc808\ub300\uc5f0\uc18d<\/span>(absolutely continuous)\uc774 \uc788\ub2e4. \uc2e4\ud568\uc218 \\(f\\colon[a,b]\\to\\mathbb R\\)\uac00 \uc808\ub300\uc5f0\uc18d\uc774\ub77c\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud574 \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc11c\ub85c \uacb9\uce58\uc9c0 \uc54a\ub294 \uc720\ud55c \uac1c\uc758 \uad6c\uac04 \\((a_i,b_i)\\subseteq[a,b]\\)\uac00<br \/>\n\\[<br \/>\n\\sum_i(b_i-a_i)&lt;\\delta<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0ac \ub54c\ub9c8\ub2e4<br \/>\n\\[<br \/>\n\\sum_i|f(b_i)-f(a_i)|&lt;\\varepsilon<br \/>\n\\]<br \/>\n\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 4.26.<\/span><br \/>\n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f(x)=\\sqrt x\\)\ub294 \\([0,1]\\)\uc5d0\uc11c \\(1\/2\\)-\ud694\ub354 \uc5f0\uc18d\uc774\uc9c0\ub9cc \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc740 \uc544\ub2c8\ub2e4.<\/li>\n<li>\\(M&gt;0\\)\uc77c \ub54c \\(f(x)=x^2\\)\uc740 \\([-M,M]\\)\uc5d0\uc11c \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc774\uace0, \ub9bd\uc2dc\uce20 \uc0c1\uc218\ub85c \\(2M\\)\uc744 \ud0dd\ud560 \uc218 \uc788\ub2e4. \ud55c\ud3b8 \\(\\mathbb R\\) \uc804\uccb4\uc5d0\uc11c\ub294 \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc774 \uc544\ub2c8\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.27.<\/span><br \/>\n\\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(f\\colon X\\to\\mathbb R\\)\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \\(a\\in X\\)\uc5d0\uc11c <span class=\"defined\">\uc0c1\ubc18\uc5f0\uc18d<\/span>(upper semicontinuous)\uc774\ub77c\ub294 \uac83\uc740 \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(d_X(x,a)&lt;\\delta\\)\uc774\uba74 \\(f(x)&lt;f(a)+\\varepsilon\\)\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4. \uac19\uc740 \ubc29\ubc95\uc73c\ub85c <span class=\"defined\">\ud558\ubc18\uc5f0\uc18d<\/span>(lower semicontinuous)\uc744 \\(f(x)&gt;f(a)-\\varepsilon\\)\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \\(f\\)\uac00 \\(a\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(a\\)\uc5d0\uc11c \uc0c1\ubc18\uc5f0\uc18d\uc774\uba74\uc11c \ud558\ubc18\uc5f0\uc18d\uc778 \uac83\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.28.<\/span><br \/>\n\\(X\\)\uc640 \\(Y\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(A\\subseteq X\\)\uc774\uba70 \\(f\\colon X\\to Y\\)\uac00 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c<br \/>\n\\[<br \/>\nf(\\overline A)\\subseteq\\overline{f(A)}<br \/>\n\\]<br \/>\n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774\ub97c \uc774\uc6a9\ud558\uc5ec \\(A\\)\uac00 \\(X\\)\uc5d0\uc11c \uc870\ubc00\ud558\uace0 \ub450 \uc5f0\uc18d\ud568\uc218 \\(f,g\\colon X\\to Y\\)\uac00 \\(A\\)\uc5d0\uc11c \uc77c\uce58\ud558\uba74 \\(X\\) \uc804\uccb4\uc5d0\uc11c \uc77c\uce58\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.29.<\/span><br \/>\n\uc704\uc0c1\uacf5\uac04 \\((X,\\mathcal T_X)\\), \\((Y,\\mathcal T_Y)\\) \uc0ac\uc774\uc758 \ud568\uc218 \\(f\\colon X\\to Y\\)\uac00 \uc5f0\uc18d\uc774\ub77c\ub294 \uac83\uc740 \ubaa8\ub4e0 \\(V\\in\\mathcal T_Y\\)\uc5d0 \ub300\ud558\uc5ec \\(f^{-1}(V)\\in\\mathcal T_X\\)\uc778 \uac83\uc744 \ub73b\ud55c\ub2e4. \uc77c\ub300\uc77c\ub300\uc751 \\(f\\colon X\\to Y\\)\uc640 \uc5ed\ud568\uc218 \\(f^{-1}\\)\uac00 \ubaa8\ub450 \uc5f0\uc18d\uc774\uba74 \\(f\\)\ub97c <span class=\"defined\">\uc704\uc0c1\ub3d9\ud615<\/span>(homeomorphism)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uac70\ub9ac\uacf5\uac04\uc5d0 \uac70\ub9ac\uc5d0\uc11c \uc720\ub3c4\ub41c \uc704\uc0c1\uc744 \uc8fc\uba74 \uc774 \uc5f0\uc18d\uc131\uc758 \uc815\uc758\uac00 \uc815\ub9ac 4.5\uc758 \uc815\uc758\uc640 \uc77c\uce58\ud568\uc744 \uc124\uba85\ud558\uc2dc\uc624. \ub610\ud55c \uc704\ub85c\uc758 \uac70\ub9ac\ubcf4\uc874\ud568\uc218\ub294 \uc704\uc0c1\ub3d9\ud615\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.30.<\/span><br \/>\n\\(a&lt;b\\)\uc774\uace0 \\(C[a,b]\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \\(\\mathbb R\\)\ub85c\uc758 \uc5f0\uc18d\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(C[a,b]\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\uac00 \\(\\mathbb R\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\uc640 \uac19\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624. \\(\\mathbb Q\\cap[a,b]\\)\uc5d0\uc11c\uc758 \ud568\uc22b\uac12\ub4e4\uc774 \uc5f0\uc18d\ud568\uc218\ub97c \uc720\uc77c\ud558\uac8c \uacb0\uc815\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 4.31.<\/span><br \/>\n\ud568\uc218 \\(f\\colon\\mathbb R\\to\\mathbb R\\)\uac00 \ubaa8\ub4e0 \uc2e4\uc218 \\(s,t\\)\uc5d0 \ub300\ud558\uc5ec \\(f(s+t)=f(s)+f(t)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f\\)\uac00 \uc5f0\uc18d\uc778 \uc810\uc774 \ud558\ub098 \uc774\uc0c1 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \uc2e4\uc218 \\(a\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x)=ax\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(f\\)\uac00 \uc5f0\uc18d\uc778 \uc810\uc774 \ud558\ub098\ub3c4 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc744 \uc218 \uc788\ub294\uac00? \uc774 \ubb38\ud56d\uc5d0\uc11c\ub294 \\(\\mathbb R\\)\uac00 \\(\\mathbb Q\\) \uc704\uc758 \ubca1\ud130\uacf5\uac04\uc73c\ub85c\uc11c \ud558\uba5c \uae30\uc800\ub97c \uac00\uc9c4\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/\">\ud574\uc11d\ud559 \uac15\uc758\ub178\ud2b8<\/a><\/p>\n<ol class=\"contentboxorderedlist\">\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch01-real-number-system\">\uc2e4\uc218\uacc4\uc758 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch02-metric-spaces\">\uac70\ub9ac\uacf5\uac04<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch03-limit-of-sequences\">\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uc704\uc0c1\uc801 \uc131\uc9c8<\/a><\/li>\n<li class=\"contentboxthis\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch04-limit-of-functions-and-continuity\">\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch05-differentiation-of-functions-of-one-variable\">\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch06-the-riemann-integral\">\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch07-infinite-series\">\ubb34\ud55c\uae09\uc218<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch08-real-analytic-functions\">\uc2e4\ud574\uc11d\uc801 \ud568\uc218<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\">\ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\">\uc911\uc801\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch11-vector-field-and-fundamental-theorems\">\ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><!-- ################## --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc55e \uc7a5\uc5d0\uc11c\ub294 \uac70\ub9ac\uacf5\uac04\uc5d0\uc11c \uc218\uc5f4\uc758 \uc218\ub834, \uc5f4\ub9b0\uc9d1\ud569\uacfc \ub2eb\ud78c\uc9d1\ud569, \ucef4\ud329\ud2b8\uc131\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \uac1c\ub150\uc744 \ubc14\ud0d5\uc73c\ub85c \uac70\ub9ac\uacf5\uac04 \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131\uc744 \ub2e4\ub8ec\ub2e4. \ud568\uc218\uc758 \uadf9\ud55c \uac70\ub9ac\uacf5\uac04 \\((X,d_X)\\), \\((Y,d_Y)\\)\uc640 \ud568\uc218 \\(f\\colon X\\to Y\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc810 \\(a\\in X\\)\uac00 \\(X\\)\uc758 \uc9d1\uc801\uc810\uc774\uace0 \\(L\\in Y\\)\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud574 \\(\\delta&gt;0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(0&lt;d_X(x,a)&lt;\\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(d_Y(f(x),L)&lt;\\varepsilon\\)\uc774 \uc131\ub9bd\ud558\uba74, \u201c\\(x\\to a\\)\uc77c \ub54c \\(f(x)\\)\uac00 \\(L\\)\ub85c \uc218\ub834\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\uac83\uc744 \uae30\ud638\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \\(&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9470,"menu_order":104,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9484","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9484","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"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=9484"}],"version-history":[{"count":14,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9484\/revisions"}],"predecessor-version":[{"id":10108,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9484\/revisions\/10108"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9470"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9484"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}