{"id":6703,"date":"2021-07-20T23:59:48","date_gmt":"2021-07-20T14:59:48","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6703"},"modified":"2021-09-21T20:44:21","modified_gmt":"2021-09-21T11:44:21","slug":"properties-of-continuous-functions","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/properties-of-continuous-functions\/","title":{"rendered":"\uc5f0\uc18d\ud568\uc218\uc758 \uc131\uc9c8"},"content":{"rendered":"<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 4\uc7a5 2\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\uc815\ub9ac 3.2.1\uc5d0\uc11c \uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4 \ubcf4\uc558\ub2e4. \ud568\uc218\uc758 \uc5f0\uc18d\uc131\uc774 \ud568\uc218\uc758 \uadf9\ud55c\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ub418\ubbc0\ub85c, \ud568\uc218\uc758 \uc5f0\uc18d\uc131 \ub610\ud55c \uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ubc00\uc811\ud558\uac8c \uad00\ub828\ub418\uc5b4 \uc788\ub2e4. \ub2e4\uc74c \uc815\ub9ac\ub97c \uc0b4\ud3b4 \ubcf4\uc790.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 4.2.1. (\uc218\uc5f4 \uc5f0\uc18d, \uc810\uc5f4 \uc5f0\uc18d, Sequential Continuity)<\/span><\/p>\n<p>\\(f:D \\rightarrow \\mathbb{R}\\)\uac00 \ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\uba74\uc11c \\(D\\)\uc758 \uc6d0\uc18c\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(c\\)\uc5d0\uc11c \\(f\\)\uac00 \uc5f0\uc18d\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ub450 \uc870\uac74<\/p>\n<ul>\n<li>\\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(x_n \\rightarrow c,\\)<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\in D\\)<\/li>\n<\/ul>\n<p>\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubaa8\ub4e0 \uc218\uc5f4 \\(\\left\\{x_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \uc218\uc5f4 \\(\\left\\{ f(x_n ) \\right\\}\\)\uc774 \\(f(c)\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc218\uc5f4 \uc5f0\uc18d\uc744 \uc774\uc6a9\ud558\uba74 \ud568\uc218\uc758 \uadf9\ud55c\uc774 \uc218\ub834\ud558\uc9c0 \uc54a\uc74c\uc744 \uc27d\uac8c \ubcf4\uc77c \uc218 \uc788\ub2e4. \ub2e4\uc74c \uc608\ub97c \ubcf4\uc790.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.2.2.<\/span><br \/>\n\ud568\uc218 \\(f:D \\rightarrow \\mathbb{R}\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[f(x) = \\begin{cases}<br \/>\n\\displaystyle\\sin\\frac{1}{x} &#038; \\quad \\text{if} \\,\\, x\\ne 0 , \\\\[4pt]<br \/>\n0 &#038; \\quad \\text{if} \\,\\, x=0 .<br \/>\n\\end{cases}\\]<br \/>\n\uc218\uc5f4 \\(\\left\\{ s_n \\right\\}\\)\uacfc \\(\\left\\{ t_n \\right\\}\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<br \/>\n\\[s_n = \\frac{1}{n\\pi} ,\\quad \\, t_n = \\frac{2}{(4n+1)\\pi} .\\]<br \/>\n\uadf8\ub7ec\uba74 \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(s_n \\rightarrow 0\\)\uc774\uace0 \\(t_n \\rightarrow 0\\)\uc774\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[\\lim_{n\\rightarrow\\infty} f(s_n) = \\lim_{n\\rightarrow\\infty} \\sin(n\\pi ) = \\lim_{n\\rightarrow\\infty} 0 =0\\]<br \/>\n\uc774\uace0<br \/>\n\\[\\lim_{n\\rightarrow\\infty} f(t_n) = \\lim_{n\\rightarrow\\infty} \\sin \\frac{(4n+1)\\pi}{2} = \\lim_{n\\rightarrow\\infty}\\sin\\left( 2n\\pi + \\frac{\\pi}{2}\\right) = \\lim_{n\\rightarrow\\infty} 1 = 1\\]<br \/>\n\uc774\ub2e4. \ub9cc\uc57d \\(f\\)\uac00 \\(0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uba74 \uc774 \ub450 \uadf9\ud55c\uc774 \uc77c\uce58\ud574\uc57c \ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(0\\ne 1\\)\uc774\ubbc0\ub85c, \\(f\\)\ub294 \\(0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2c8\ub2e4.\n<\/p>\n<\/div>\n<p>\ub2e4\uc74c\uc740 \ud568\uc218\uac00 \ubd88\uc5f0\uc18d\uc774\uae30 \ub54c\ubb38\uc5d0 \ubc1c\uc0dd\ud558\ub294 \ud604\uc0c1\uc744 \uc124\uba85\ud55c \uac83\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.2.3.<\/span><\/p>\n<p>\ud568\uc218 \\(f(x) = \\lfloor x \\rfloor\\)\ub77c\uace0 \ud558\uace0<br \/>\n\\[x_n = \\sum_{k=1}^{n} \\frac{9}{10^k}\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\lim_{n\\rightarrow\\infty} = \\sum_{k=1}^{\\infty} \\frac{9}{10^k} = 0.999 \\cdots = 1\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[f\\left(\\lim_{n\\rightarrow\\infty} x_n \\right) = f(0.999 \\cdots ) = \\lfloor 0.999 \\cdots \\rfloor = \\lfloor 1 \\rfloor = 1\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ub098<br \/>\n\\[\\lim_{n\\rightarrow\\infty} f( x_n ) = \\lim_{n\\rightarrow\\infty} \\left\\lfloor \\sum_{k=1}^n \\frac{9}{10^k} \\right\\rfloor = \\lim_{n\\rightarrow\\infty} 0 = 0\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\lfloor 0.999 \\cdots \\rfloor \\ne \\lim_{n\\rightarrow\\infty} \\lfloor \\,\\underbrace{0.999 \\cdots 9}_{n\\,\\,\\text{times}} \\,\\rfloor \\]<br \/>\n\uc774\ub2e4. \uc591\ubcc0\uc774 \uc77c\uce58\ud558\uc9c0 \uc54a\ub294 \uc774\uc720\ub294 \ucd5c\ub300\uc815\uc218\ud568\uc218\uac00 \\(1\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2c8\uae30 \ub54c\ubb38\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc815\ub9ac 4.2.1\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c \ub450 \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 4.2.2. (\ud569\uc131\ud568\uc218\uc758 \uadf9\ud55c)<\/span><\/p>\n<p>\ud568\uc218 \\(f:A \\rightarrow B\\)\uc640 \\(g:B\\rightarrow C\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(c\\)\uac00 \\(A\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(x\\rightarrow c\\)\uc77c \ub54c \\(f(x)\\)\uac00 \\(L\\)\uc5d0 \uc218\ub834\ud558\uace0, \\(L\\in B\\)\uc774\uba70 \\(g\\)\uac00 \\(L\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\lim_{x\\rightarrow c} g(f(x)) = g\\left( \\lim_{x\\rightarrow c} f(x)\\right) = g(L).\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 4.2.3. (\ud569\uc131\ud568\uc218\uc758 \uc5f0\uc18d\uc131)<\/span><\/p>\n<p>\ud568\uc218 \\(f:A \\rightarrow B\\)\uc640 \\(g:B\\rightarrow C\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(c\\)\uac00 \\(A\\)\uc758 \uc9d1\uc801\uc810\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(g\\)\uac00 \\(g(c)\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \ud569\uc131\ud568\uc218 \\(g\\circ f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.2.4.<\/span><br \/>\n\\(n\\)\uc774 \uc591\uc758 \uc815\uc218\uc774\uace0 \\(f(x)=x^n\\)\uc774\uba70 \\(g(x)=\\sin x\\)\ub77c\uace0 \ud558\uc790. \\(f\\)\uc640 \\(g\\)\uac00 \ubaa8\ub450 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c<br \/>\n\\[(f\\circ g)(x) = \\sin^n x\\]<br \/>\n\ub85c \uc8fc\uc5b4\uc9c4 \ud569\uc131\ud568\uc218 \\(f\\circ g\\) \ub610\ud55c \uc5f0\uc18d\ud568\uc218\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.2.5.<\/span><br \/>\n\\(f(x) = \\lfloor x \\rfloor\\)\uc774\uace0 \\(g(x)=x^2\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\\(g\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(f\\)\uac00 \uc815\uc218\uac00 \uc544\ub2cc \uc810\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \ud569\uc131\ud568\uc218 \\(g\\circ f\\)\uac00 \uc815\uc218\uac00 \uc544\ub2cc \uc810\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(n\\)\uc774 \uc815\uc218\ub77c\uba74<br \/>\n\\[\\lim_{x\\rightarrow n^-} (g\\circ f)(x) = g\\left(\\lim_{x\\rightarrow n^-} f(x)\\right) = (n-1)^2\\]<br \/>\n\uc774\uace0<br \/>\n\\[\\lim_{x\\rightarrow n^+} (g\\circ f)(x) = g\\left(\\lim_{x\\rightarrow n^+} f(x) \\right) = n^2\\]<br \/>\n\uc774\ub2e4. \\((n-1)^2 \\ne n^2\\)\uc774\ubbc0\ub85c, \\(x\\rightarrow n\\)\uc77c \ub54c \\(g\\circ f\\)\uac00 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(g\\circ f\\)\uac00 \\(n\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n\n\n\n<p>.<\/p>\n\n\n\n\n\n<p>.<\/p>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/continuity\">\uc5f0\uc18d\uc758 \uc815\uc758<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/theorems-on-continuity\">\uc5f0\uc18d\uc131\uacfc \uad00\ub828\ub41c \uc815\ub9ac<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 4\uc7a5 2\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \uc815\ub9ac 3.2.1\uc5d0\uc11c \uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uad00\uacc4\ub97c \uc0b4\ud3b4 \ubcf4\uc558\ub2e4. \ud568\uc218\uc758 \uc5f0\uc18d\uc131\uc774 \ud568\uc218\uc758 \uadf9\ud55c\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ub418\ubbc0\ub85c, \ud568\uc218\uc758 \uc5f0\uc18d\uc131 \ub610\ud55c \uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \ubc00\uc811\ud558\uac8c \uad00\ub828\ub418\uc5b4 \uc788\ub2e4. \ub2e4\uc74c \uc815\ub9ac\ub97c \uc0b4\ud3b4 \ubcf4\uc790. \uc815\ub9ac 4.2.1. (\uc218\uc5f4 \uc5f0\uc18d, \uc810\uc5f4 \uc5f0\uc18d, Sequential Continuity) \\(f:D \\rightarrow \\mathbb{R}\\)\uac00 \ud568\uc218\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \uc9d1\uc801\uc810\uc774\uba74\uc11c \\(D\\)\uc758 \uc6d0\uc18c\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(c\\)\uc5d0\uc11c \\(f\\)\uac00 \uc5f0\uc18d\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ub450 \uc870\uac74&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":402,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6703","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6703","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=6703"}],"version-history":[{"count":20,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6703\/revisions"}],"predecessor-version":[{"id":7869,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6703\/revisions\/7869"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}