{"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":"2025-10-21T16:08:05","modified_gmt":"2025-10-21T07:08:05","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>\uc774 \uc7a5\uc5d0\uc11c\ub294 \uac70\ub9ac\uacf5\uac04 \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131\uc744 \uc0b4\ud3b4\ubcf8\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: X \\to Y\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc810 \\(a \\in X\\)\uac00 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc9d1\uc801\uc810\uc774\uace0 \\(L\\)\uc774 \\(f\\)\uc758 \uacf5\uc5ed\uc758 \uc810\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574 \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(0 < d_X(x,\\, a) < \\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(d_Y(f(x),\\, L) < \\varepsilon\\)\uc774 \uc131\ub9bd\ud558\uba74, \"\\(x \\to a\\)\uc77c \ub54c \\(f(x)\\)\uac00 \\(L\\)\ub85c <span class=\"defined\">\uc218\ub834<\/span>\ud55c\ub2e4&#8221;\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\uac83\uc744 \uae30\ud638\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[\\lim_{x \\to a} f(x) = L \\quad \\text{ \ub610\ub294 } \\quad f(x) \\to L \\,\\,\\text{ as }\\,\\, x \\to a.\\]<\/p>\n<p>\uc218\uc5f4\uc744 \uc0ac\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uadf9\ud55c\uc744 \uc815\uc758\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 4.1. (\ud558\uc774\ub124 \uc815\ub9ac)<\/span><\/p>\n<p>\\(X\\)\uc640 \\(Y\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(f:X\\rightarrow 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=\"proofname\">\uc99d\uba85<\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \\(L\\)\uc5d0 \uc218\ub834\ud560 \ub54c, \uc870\uac74 (2)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc784\uc758\uc758 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x_n) \\rightarrow L\\)\uc774 \uc131\ub9bd\ud568\uc740 \uc790\uba85\ud558\ub2e4.<\/p>\n<p>\uc5ed\uc744 \uc99d\uba85\ud558\uc790. \uc989 \uc870\uac74 (2)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc784\uc758\uc758 \uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x_n) \\rightarrow L\\)\uc774 \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \uacb0\ub860\uacfc\ub294 \ubc18\ub300\ub85c \\(x\\rightarrow a\\)\uc77c \ub54c \\(f(x)\\)\uac00 \\(L\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(\\varepsilon_0 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(\\delta = \\frac{1}{n}\\)\uc5d0 \ub300\ud558\uc5ec \\(0 < d_X(x_n,\\, a) < \\frac{1}{n}\\)\uc774\uc9c0\ub9cc \\(d_Y(f(x_n),\\, L) \\geq \\varepsilon_0\\)\uc778 \\(x_n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(\\left\\{ x_n \\right\\}\\)\uc740 \\(a\\)\uc5d0 \uc218\ub834\ud558\uc9c0\ub9cc \\(\\left\\{ f(x_n) \\right\\}\\)\uc740 \\(L\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \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\\rightarrow a\\)\uc77c \ub54c \\(f(x)\\)\uc758 \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc74c\uc744 \ubcf4\uc774\uae30 \uc704\ud574, \\(s_n \\rightarrow a\\), \\(t_n \\rightarrow a\\)\uc774\uc9c0\ub9cc \\(\\left\\{ f(s_n )\\right\\}\\)\uacfc \\(\\left\\{ f(t_n )\\right\\}\\)\uc774 \ub2e4\ub978 \uac12\uc5d0 \uc218\ub834\ud558\ub294 \ub450 \uc218\uc5f4 \\(\\left\\{ s_n \\right\\}\\), \\(\\left\\{ t_n \\right\\}\\)\uc744 \ucc3e\uc73c\uba74 \ub41c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.1.<\/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}:X\\rightarrow\\left\\{ 0,\\,1\\right\\}\\)\uc740 \\(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 \\(\\left\\{ s_n \\right\\}\\)\uacfc \\(a\\)\ub85c \uc218\ub834\ud558\uace0 \\(t_n \\neq a\\)\uc778 \ubb34\ub9ac\uc218\uc5f4 \\(\\left\\{ t_n \\right\\}\\)\uc744 \ud0dd\ud558\uba74 \\(f ( s_n ) \\rightarrow 1\\)\uc774\uc9c0\ub9cc \\(f( t_n )\\rightarrow 0\\)\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 limits)\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc989 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(X\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 \\(f:X \\rightarrow Y\\)\uc640 \uc810 \\(a\\), \\(L\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(f\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(X\\cap (-\\infty, a)\\)\ub85c \uc81c\ud55c\ud588\uc744 \ub54c \\(x\\rightarrow a\\)\uc778 \uadf9\ud55c\uc744 \\(f\\)\uc758 <span class=\"defined\">\uc88c\uadf9\ud55c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(0 < a-x < \\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x)-L| < \\varepsilon\\)\uc774 \uc131\ub9bd\ud558\uba74, \"\\(x\\to a-\\)\uc77c \ub54c \\(f(x)\\)\uac00 \\(L\\)\ub85c \uc218\ub834\ud55c\ub2e4\"\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \\(\\displaystyle \\lim_{x\\to a-} f(x) = L\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<li>\\(f\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(X\\cap (a, \\infty)\\)\ub85c \uc81c\ud55c\ud588\uc744 \ub54c \\(x\\rightarrow a\\)\uc778 \uadf9\ud55c\uc744 \\(f\\)\uc758 <span class=\"defined\">\uc6b0\uadf9\ud55c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(0 < x-a < \\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x)-L| < \\varepsilon\\)\uc774 \uc131\ub9bd\ud558\uba74, \"\\(x\\to a+\\)\uc77c \ub54c \\(f(x)\\)\uac00 \\(L\\)\ub85c \uc218\ub834\ud55c\ub2e4\"\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \\(\\displaystyle \\lim_{x\\to a+} f(x) = L\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/li>\n<\/ul>\n<p>\uc2e4\ud568\uc218 \\(f\\)\uac00 \uc810 \\(a\\)\uc5d0\uc11c \uadf9\ud55c\uc744 \uac00\uc9c8 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc774 \ubaa8\ub450 \uc874\uc7ac\ud558\uace0 \uac19\uc740 \uac83\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.1.<\/span><br \/>\n\uacf5\uc5ed\uc774 \\(\\mathbb{R}\\)\uc778 \ud568\uc218\uc758 \uadf9\ud55c\uc5d0\uc11c \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uadf9\ud55c, \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uadf9\ud55c\uc758 \uc815\uc758\ub97c \uc870\uc0ac\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:D\\rightarrow\\mathbb{R}\\)\uc640 \\(g:D\\rightarrow\\mathbb{R}\\)\uac00 \\(x\\rightarrow 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\\rightarrow c\\)\uc77c \ub54c \\((kf)(x) \\rightarrow kA\\)\uc774\ub2e4.<\/li>\n<li>\\(x\\rightarrow c\\)\uc77c \ub54c \\((f+g)(x) \\rightarrow A+B\\)\uc774\ub2e4.<\/li>\n<li>\\(x\\rightarrow c\\)\uc77c \ub54c \\((fg)(x) \\rightarrow AB\\)\uc774\ub2e4.<\/li>\n<li>\\(B\\ne 0\\)\uc774\uba74, \\(x\\rightarrow c\\)\uc77c \ub54c \\((f\/g)(x) \\rightarrow 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:X\\rightarrow\\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\\[f(x) = (f_1 (x) ,\\, f_2 (x) ,\\, f_3 (x) ,\\, \\cdots ,\\, f_d (x))\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(L=(L_1 ,\\, L_2 ,\\, \\cdots ,\\, L_d )\\in\\mathbb{R}^d\\)\ub77c\uace0 \ud558\uc790. \\(x\\rightarrow a\\)\uc77c \ub54c \\(f(x)\\rightarrow L\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(j=1,\\,2,\\,\\cdots,\\,d\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\rightarrow a\\)\uc77c \ub54c \\(f_j (x) \\rightarrow 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)\\rightarrow (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^2 y}{x^2 + y^2}\\)<\/li>\n<\/ol>\n<\/div>\n<h3>\uc5f0\uc18d\ud568\uc218<\/h3>\n<p>\ud568\uc218 \\(f: 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 > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\( d_X (x,\\,a) < \\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(d_Y ( f(x) ,\\, f(a))<\\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \ub610\ud55c \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 4.2. (\uc218\uc5f4\uc744 \uc0ac\uc6a9\ud55c \uc5f0\uc18d\uc758 \uc815\uc758)<\/span><\/p>\n<p>\ud568\uc218 \\(f:X\\rightarrow 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=\"proofname\">\uc99d\uba85<\/p>\n<p>(1)\uacfc (3)\uc774 \ub3d9\uce58\uc784\uc740 \uc5f0\uc18d\uc758 \uc815\uc758\uc640 \uadf9\ud55c\uc758 \uc815\uc758\ub85c\ubd80\ud130 \uace7\ubc14\ub85c \uc5bb\ub294\ub2e4. \ub610\ud55c (2)\uc640 (3)\uc774 \ub3d9\uce58\uc784\uc740 \ud558\uc774\ub124 \uc815\ub9ac(\uc815\ub9ac 4.1)\ub85c\ubd80\ud130 \uc54c \uc218 \uc788\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:I\\rightarrow\\mathbb{R}\\)\uc774 \ub2e8\uc870\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f\\)\uc758 \uc88c\uadf9\ud55c\uacfc \uc6b0\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.<\/p>\n<\/div>\n<p>\uc790\uc8fc \ub9cc\ub098\ub294 \uc5f0\uc18d\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li>\ud56d\ub4f1\ud568\uc218 \\(\\operatorname{id}_X : X \\to X\\)\ub294 \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\uc0c1\uc218\ud568\uc218\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4.<\/li>\n<li>\uc5f0\uc18d\ud568\uc218\ub97c \ud569\uc131\ud558\uc5ec \ub9cc\ub4e0 \ud568\uc218\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4. \uc989, \\(f: X \\to Y\\)\uc640 \\(g: Y \\to Z\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uba74 \\(g \\circ f: X \\to Z\\)\ub3c4 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4.<\/li>\n<li>\uc0ac\uc601\ud568\uc218 \\(\\pi_i: X_1 \\times X_2 \\to X_i\\)\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4.<\/li>\n<li>\\(\\mathbb{R}^n\\)\uc5d0\uc11c \\(\\mathbb{R}\\)\ub85c\uc758 \ub178\ub984 \\(x \\mapsto \\|x\\|\\)\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4.<\/li>\n<li>\uac70\ub9ac\ud568\uc218 \\(d: X \\times X \\to \\mathbb{R}\\)\uc740 \uc5f0\uc18d\ud568\uc218\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(x)\\ne 0\\)\uc778 \ubaa8\ub4e0 \uc810 \\(x\\)\uc5d0\uc11c \\(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:X\\rightarrow Y\\)\uc640 \\(g:Y\\rightarrow Z\\)\uac00 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(a\\in X&#8217;\\)\uc774\uba70, \\(x\\rightarrow a\\)\uc77c \ub54c \\(f(x) \\rightarrow 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\\[\\lim_{x\\rightarrow a} g(f(x)) = g \\left( \\lim_{x\\rightarrow a} f(x) \\right).\\]<\/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>0\\)\uc77c \ub54c \\(\\ln x = \\log_e x\\)\uc774\ub2e4. \ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\lim_{x\\rightarrow \\infty} \\left(1+\\frac{1}{x}\\right)^{x}\\)<\/li>\n<li>\\(\\lim_{x\\rightarrow 0} (1+x)^{\\frac{1}{x}}\\)<\/li>\n<li>\\(\\lim_{x\\rightarrow 0} \\frac{\\ln (1+x)}{x}\\)<\/li>\n<li>\\(\\lim_{x\\rightarrow 0} \\frac{e^x -1}{x}\\)<\/li>\n<li>\\(\\lim_{x\\rightarrow 0} \\frac{\\log_a (1+x)}{x}\\) (\ub2e8, \\(a>0\\), \\(a\\ne 1\\).)<\/li>\n<li>\\(\\lim_{x\\rightarrow 0} \\frac{a^x -1}{x}\\) (\ub2e8, \\(a>0\\), \\(a\\ne 1\\).)<\/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>\\(\\lim_{x\\rightarrow 0} \\frac{\\sin x}{x}\\)<\/li>\n<li>\\(\\lim_{x\\rightarrow 0} \\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 4.3. (\uc5f4\ub9b0\uc9d1\ud569\uc744 \uc0ac\uc6a9\ud55c \uc5f0\uc18d\uc758 \uc815\uc758)<\/span><\/p>\n<p>\ud568\uc218 \\(f: X \\to Y\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc77c \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=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(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\ub2e4. \\(V\\)\uac00 \uc5f4\ub9b0\uc9d1\ud569\uc774\ubbc0\ub85c \\(\\varepsilon > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(B(f(x),\\, \\varepsilon) \\subseteq V\\)\uc774\ub2e4. \\(f\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(B(x,\\, \\delta) \\subseteq f^{-1}(V)\\)\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(Y\\)\uc758 \ubaa8\ub4e0 \uc5f4\ub9b0\uc9d1\ud569 \\(V\\)\uc5d0 \ub300\ud574 \\(f^{-1}(V)\\)\uac00 \\(X\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(a\\in X\\)\uc640 \\(\\varepsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(V=B(f(a),\\,\\epsilon)\\)\uc774 \\(Y\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc774\ubbc0\ub85c \\(f\\)\uc5d0 \uc758\ud55c \\(V\\)\uc758 \uc5ed\uc0c1\uc774 \\(X\\)\uc758 \uc5f4\ub9b0\uc9d1\ud569\uc774\ub2e4. \ub610\ud55c \\(a\\in X\\)\uc774\ubbc0\ub85c \\(B(a,\\,\\delta)\\subseteq f^{-1}(V)\\)\uc778 \\(\\delta >0\\)\uc774 \uc874\uc7ac\ud55c\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 4.4. (\ub2eb\ud78c\uc9d1\ud569\uacfc \uc5f0\uc18d\ud568\uc218\uc758 \uad00\uacc4)<\/span><\/p>\n<p>\ud568\uc218 \\(f:X\\rightarrow 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 4.4)\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:X\\rightarrow Y\\)\uc640 \\(g:Y\\rightarrow Z\\)\uac00 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ud569\uc131\ud568\uc218 \\(g\\circ f : X\\rightarrow 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:\\mathbb{R}\\times\\mathbb{R} \\rightarrow \\mathbb{R}\\)\uc774 \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) = \\lvert x-y \\rvert\\)<\/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\\)\uc640 \\(g\\)\uac00 \\(X\\)\ub85c\ubd80\ud130 \\(\\mathbb{R}\\)\ub85c\uc758 \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\uc640 \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:X\\times X \\rightarrow \\mathbb{R}\\)\uc774 \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\\)\uc640 \\(Y\\)\uac00 \ucef4\ud329\ud2b8 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \ud568\uc218 \\(f:X\\rightarrow Y\\)\uac00 \uc77c\ub300\uc77c\ub300\uc751\uc774\uba70 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \uc5ed\ud568\uc218 \\(f^{-1} : Y \\rightarrow X\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\ud2b9\ud788 \\(Y\\subseteq\\mathbb{R}\\)\uc77c \ub54c \ub2e4\uc74c \uc815\ub9ac\uac00 \uc720\uc6a9\ud558\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 4.5. (\ucd5c\ub300 \ucd5c\uc18c \uc815\ub9ac)<\/span><\/p>\n<p>\\(f:X\\rightarrow \\mathbb{R}\\)\uc774 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(K\\)\uac00 \\(X\\)\uc758 \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=\"proofname\">\uc99d\uba85<\/p>\n<p>\ud568\uc218 \\(f: K \\to \\mathbb{R}\\)\uc774 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(K\\)\uac00 \ucef4\ud329\ud2b8\uc774\uba74 \\(f(K)\\)\ub3c4 \ucef4\ud329\ud2b8\uc774\ub2e4. \\(\\mathbb{R}\\)\uc758 \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569\uc740 \ub2eb\ud600\uc788\uace0 \uc720\uacc4\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f(K)\\)\uc758 \uc0c1\ud55c\uacfc \ud558\ud55c\uc774 \uc874\uc7ac\ud558\uace0, \\(f(K)\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc774\ubbc0\ub85c \\(f(K)\\)\uc758 \uc0c1\ud55c\uacfc \ud558\ud55c\uc774 \\(f(K)\\)\uc5d0 \uc18d\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uac00 <span class=\"defined\">\uc5f0\uacb0<\/span>\ub418\uc5b4 \uc788\ub2e4(connected)\ub294 \uac83\uc740, \\(X\\)\ub97c \ub450 \uac1c\uc758 \ube44\uc5b4\uc788\uc9c0 \uc54a\uc740 \uc5f4\ub9b0\uc9d1\ud569\uc758 \ubd84\ub9ac\ub41c \ud569\uc9d1\ud569\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc5c6\ub294 \uac83\uc774\ub2e4. \uc989, \\(X = U \\cup V\\)\uc774\uace0 \\(U \\cap V = \\varnothing\\)\uc778 \ube44\uc5b4\uc788\uc9c0 \uc54a\uc740 \uc5f4\ub9b0\uc9d1\ud569 \\(U,\\, V\\)\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\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\uc77c \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<p>\uac70\ub9ac\uacf5\uac04 \\(X\\)\uac00 <span class=\"defined\">\uacbd\ub85c\uc5f0\uacb0<\/span>\ub418\uc5b4 \uc788\ub2e4(path-connected)\ub294 \uac83\uc740, \uc784\uc758\uc758 \ub450 \uc810 \\(x,\\, y \\in X\\)\uc5d0 \ub300\ud574 \uc5f0\uc18d\ud568\uc218 \\(\\gamma: [0,\\, 1] \\to X\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\gamma(0) = x\\)\uc774\uace0 \\(\\gamma(1) = y\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.16.<\/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.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.17.<\/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\\[A = \\left\\{ (0,\\,y) \\mid y\\in\\mathbb{R}\\right\\},\\quad B = \\left\\{ (x,\\,y) \\,\\Bigg\\vert\\, y=\\sin\\frac{1}{x} ,\\, x>0\\right\\}.\\]<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<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 4.6. (\\(\\mathbb{R}\\)\uc5d0\uc11c \uc5f0\uacb0\ub41c \uc9d1\ud569\uc758 \ud615\ud0dc)<\/span><\/p>\n<p>\\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774 \uc5f0\uacb0\ub418\uc5b4 \uc788\uc744 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uad6c\uac04\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<p>\uc5f0\uacb0\uc131\uc740 \uc5f0\uc18d\ud568\uc218\uc5d0 \uc758\ud574 \ubcf4\uc874\ub41c\ub2e4. \uc989, \\(f: 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 4.7. (\uc0ac\uc787\uac12 \uc815\ub9ac)<\/span><\/p>\n<p>\ud568\uc218 \\(f: X \\to \\mathbb{R}\\)\uc774 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(X\\)\uac00 \uc5f0\uacb0\ub41c \uc9d1\ud569\uc774\uba74 \\(f(X)\\)\ub294 \uad6c\uac04\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.18.<\/span><br \/>\n\uc815\ub9ac 4.6\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.19.<\/span><br \/>\n\ud568\uc218 \\(f:X\\rightarrow Y\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(X\\)\uac00 \uc5f0\uacb0\ub41c \uc9d1\ud569\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 4.7\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uc810 \\(x \\in X\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uac00\uc7a5 \ud070 \uc5f0\uacb0\ub41c \ubd80\ubd84\uc9d1\ud569\uc744 \\(x\\)\uc758 <span class=\"defined\">\uc5f0\uacb0\uc131\ubd84<\/span>(connected component)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(x\\)\uc758 <span class=\"defined\">\uacbd\ub85c\uc5f0\uacb0\uc131\ubd84<\/span>(path component)\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.20.<\/span><br \/>\n\\(f: [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: (0,\\, 1] \\to \\mathbb{R}\\)\uc744 \\(f(x) = \\sin \\frac{1}{x}\\)\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 \\(f: X \\to X\\)\uac00 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(0 \\leq k < 1\\)\uc778 \\(k\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x,\\,y\\in X\\)\uc5d0 \ub300\ud558\uc5ec\n\\[d(f(x),\\, f(y)) \\leq k \\cdot d(x,\\, y)\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74, \\(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\uc640 \uac19\uc740 \uc131\uc9c8\uc744 \ubc14\ub098\ud750\uc758 <span class=\"defined\">\uace0\uc815\uc810 \uc815\ub9ac<\/span>(fixed point theorem)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.23.<\/span><br \/>\n\\(I\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ub2eb\ud78c\uad6c\uac04\uc774\uace0 \ud568\uc218 \\(f:I\\rightarrow\\mathbb{R}\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f\\)\uc758 \uadf8\ub798\ud504 \\(\\left\\{ (x,\\,y)\\mid y=f(x)\\right\\}\\)\uac00 \\(\\mathbb{R}^2\\)\uc5d0\uc11c \uc5f0\uacb0\ub41c \ub2eb\ud78c\uc9d1\ud569\uc778 \uac83\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624. \uc774 \uc131\uc9c8\uc744 <span class=\"defined\">\ub2eb\ud78c \uadf8\ub798\ud504 \uc815\ub9ac<\/span>(closed graph theorem)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<h3>\uade0\ub4f1\uc5f0\uc18d\uc131<\/h3>\n<p>\ud568\uc218 \\(f: 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 > 0\\)\uc5d0 \ub300\ud574 \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \\(d_X(x,\\, y) < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x,\\, y \\in X\\)\uc5d0 \ub300\ud574 \\(d_Y(f(x),\\, f(y)) < \\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) = \\frac{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) = \\frac{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 4.8. (\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:X\\rightarrow 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=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(f\\)\uac00 \\(X\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(\\varepsilon_0 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \ubaa8\ub4e0 \\(\\delta = \\frac{1}{n}\\)\uc5d0 \ub300\ud558\uc5ec \\(d_X(x_n,\\, y_n) < \\frac{1}{n}\\)\uc774\uc9c0\ub9cc \\(d_Y(f(x_n),\\, f(y_n)) \\geq \\varepsilon_0\\)\uc778 \uc810 \\(x_n\\), \\(y_n\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(X\\)\uac00 \ucef4\ud329\ud2b8 \uacf5\uac04\uc774\ubbc0\ub85c \\(\\{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}) \\to 0\\)\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})) \\to 0\\)\uc778\ub370, \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uade0\ub4f1\uc5f0\uc18d\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\\(f:X\\rightarrow Y\\)\uac00 \\(X\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc778 \ud568\uc218\uc774\uace0 \\(\\left\\{ x_n \\right\\}\\)\uc774 \\(X\\)\uc758 \ucf54\uc2dc \uc218\uc5f4\uc774\uba74 \\(\\left\\{ f(x_n)\\right\\}\\)\uc740 \\(Y\\)\uc758 \ucf54\uc2dc \uc218\uc5f4\uc774\ub2e4.<\/li>\n<li>\\(E\\)\uac00 \\(X\\)\uc758 \uc870\ubc00\ud55c \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(f:X\\rightarrow Y\\)\uac00 \\(E\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\uba74, \\(f\\)\ub294 \\(X\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc778 \ud568\uc218\ub85c \uc720\uc77c\ud558\uac8c \ud655\uc7a5\ub41c\ub2e4.<\/li>\n<li>\uade0\ub4f1\uc5f0\uc18d\uc778 \ud568\uc218\uc758 \ud569\uc131\uc740 \uade0\ub4f1\uc5f0\uc18d\ud568\uc218\uc774\ub2e4.<\/li>\n<\/ul>\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\uade0\ub4f1\uc5f0\uc18d\ud568\uc218\uc758 \uc218\uc5f4 \\(\\{f_n\\}\\)\uc774 \ud568\uc218 \\(f\\)\ub85c \uade0\ub4f1\uc218\ub834\ud558\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: 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 \\geq 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x,\\,y\\in X\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[d_Y(f(x),\\, f(y)) \\leq L \\cdot d_X(x,\\, y)\\]<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 < 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>\ud568\uc218 \\(f: X \\to Y\\)\uac00 <span class=\"defined\">\\(\\alpha\\)-\ud694\ub354 \uc5f0\uc18d<\/span>(\u03b1-H\u00f6lder continuous)\uc774\ub77c\ub294 \uac83\uc740, \uc5b4\ub5a4 \uc0c1\uc218 \\(C \\geq 0\\)\uacfc \\(0 < \\alpha \\leq 1\\)\uc778 \\(\\alpha\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x,\\,y\\in X\\)\uc5d0 \ub300\ud558\uc5ec\n\\[d_Y(f(x),\\, f(y)) \\leq C \\cdot d_X(x,\\, y)^\\alpha\\]\n\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4. \uc5ec\uae30\uc11c \ud2b9\ubcc4\ud788 \\(\\alpha = 1\\)\uc77c \ub54c\uac00 \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<p>\ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc778 \ud568\uc218\uc640 \uadf8\ub807\uc9c0 \uc54a\uc740 \ud568\uc218\uc758 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<ul>\n<li>\\(f(x) = \\sqrt{x}\\)\ub294 \\([0,\\, 1]\\)\uc5d0\uc11c \\(\\frac{1}{2}\\)-\ud694\ub354 \uc5f0\uc18d\uc774\uc9c0\ub9cc \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc774 \uc544\ub2c8\ub2e4.<\/li>\n<li>\ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \ub3c4\ud568\uc218\uac00 \uc720\uacc4\uc778 \ud568\uc218\ub294 \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\\(x\\ne 0\\)\uc77c \ub54c \\(f(x) = x \\sin\\frac{1}{x}\\)\uc774\uace0 \\(f(0) = 0\\)\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\ub294 \\([0,\\, 1]\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\uc9c0\ub9cc \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc774 \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. \uad6c\uac04 \\([a,\\, b]\\)\uc5d0\uc11c \uc2e4\ud568\uc218 \\(f\\)\uac00 <span class=\"defined\">\uc808\ub300\uc5f0\uc18d<\/span>\uc774\ub77c\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574 \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \uacb9\uce58\uc9c0 \uc54a\ub294 \uad6c\uac04\ub4e4 \\((a_i,\\, b_i)\\)\uc758 \uae38\uc774\uc758 \ud569\uc774 \\(\\delta\\)\ubcf4\ub2e4 \uc791\uc744 \ub54c\ub9c8\ub2e4 \\(\\sum |f(b_i) &#8211; f(a_i)| < \\varepsilon\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.26.<\/span><br \/>\n\\(f: [0,\\, 1] \\to \\mathbb{R}\\)\uc774 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0, \uc591\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(|f'(x)| \\leq M\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74, \\(f\\)\uac00 \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.27.<\/span><br \/>\n<span class=\"defined\">\uc0c1\ubc18\uc5f0\uc18d<\/span>(upper continuity)\uacfc <span class=\"defined\">\ud558\ubc18\uc5f0\uc18d<\/span>(lower continuity)\uc758 \uac1c\ub150\uc744 \uc870\uc0ac\ud558\uace0, \uc774 \uc5f0\uc18d\uc758 \uac1c\ub150\uc774 \uc774 \uae00\uc758 \uc55e\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \uc5f0\uc18d\uc758 \uac1c\ub150\uacfc \uc5b4\ub5a0\ud55c \uad00\uacc4\uac00 \uc788\ub294\uc9c0 \ubc1d\ud788\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:X\\rightarrow Y\\)\uac00 \uc5f0\uc18d\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f(\\overline{A})\\subseteq\\overline{f(A)}\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. (\uc774 \uc131\uc9c8\uc740 \uc5f0\uc18d\ud568\uc218\uc758 \uac12\uc774 \uc815\uc758\uc5ed\uc758 \uc870\ubc00\ud55c \ubd80\ubd84\uc9d1\ud569 \uc704\uc5d0\uc11c\uc758 \ud568\uc22b\uac12\uc5d0 \uc758\ud558\uc5ec \uc644\uc804\ud788 \uacb0\uc815\ub428\uc744 \uc758\ubbf8\ud55c\ub2e4.)<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.29.<\/span><br \/>\n\uc704\uc0c1\uacf5\uac04 \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc758 \uc5f0\uc18d\uc131\uc758 \uc815\uc758\ub97c \uc870\uc0ac\ud558\uc2dc\uc624. \ub610\ud55c \uc704\uc0c1\ub3d9\ud615\uc758 \uc815\uc758\ub97c \uc870\uc0ac\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 4.30.<\/span><br \/>\n\\(C[a,\\,b]\\)\uac00 \\([a,\\,b]\\)\ub85c\ubd80\ud130 \\(\\mathbb{R}\\)\ub85c\uc758 \uc5f0\uc18d\ud568\uc218\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.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 4.31.<\/span><br \/>\n\ud568\uc218 \\(f:\\mathbb{R}\\rightarrow\\mathbb{R}\\)\uc774 \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?<\/li>\n<\/ol>\n<\/div>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/\">\ud574\uc11d\ud559 \ud575\uc2ec\uc815\ub9ac \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<\/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>\uc774 \uc7a5\uc5d0\uc11c\ub294 \uac70\ub9ac\uacf5\uac04 \uc0ac\uc774\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ud568\uc218\uc758 \uadf9\ud55c \uac70\ub9ac\uacf5\uac04 \\((X,\\, d_X)\\), \\((Y,\\, d_Y)\\)\uc640 \ud568\uc218 \\(f: X \\to Y\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc810 \\(a \\in X\\)\uac00 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc9d1\uc801\uc810\uc774\uace0 \\(L\\)\uc774 \\(f\\)\uc758 \uacf5\uc5ed\uc758 \uc810\uc774\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574 \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(0 < d_X(x,\\, a) < \\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(d_Y(f(x),\\, L) < \\varepsilon\\)\uc774 \uc131\ub9bd\ud558\uba74, \"\\(x \\to a\\)\uc77c \ub54c \\(f(x)\\)\uac00 \\(L\\)\ub85c&hellip;\n<\/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":13,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9484\/revisions"}],"predecessor-version":[{"id":9609,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9484\/revisions\/9609"}],"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}]}}