{"id":1934,"date":"2019-03-06T11:54:03","date_gmt":"2019-03-06T02:54:03","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1934"},"modified":"2021-09-22T01:28:51","modified_gmt":"2021-09-21T16:28:51","slug":"calculus-continuity","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-continuity\/","title":{"rendered":"\uc5f0\uc18d\ud568\uc218"},"content":{"rendered":"<p>\uc9c1\uad00\uc801\uc73c\ub85c, \ud568\uc218\uac00 \uc5f0\uc18d\uc774\ub77c\ub294 \uac83\uc740 \uadf8 \uadf8\ub798\ud504\uac00 \ub04a\uc5b4\uc9c0\uc9c0 \uc54a\uace0 \uc774\uc5b4\uc838 \uc788\ub294 \uac83\uc774\ub2e4. \uadf8\ub7ec\ub098 \uc774\uc640 \uac19\uc740 \uc9c1\uad00\uc801 \uac1c\ub150\ub9cc\uc73c\ub85c\ub294 \uba85\ud655\ud558\uac8c \ub2e4\ub8f0 \uc218 \uc5c6\ub294 \uc5f0\uc18d\ud568\uc218\uc758 \uc131\uc9c8\ub4e4\uc774 \uc788\ub2e4. \uc774 \uae00\uc5d0\uc11c\ub294 \uc5f0\uc18d\uc131\uc744 \uc5c4\ubc00\ud558\uac8c \uc815\uc758\ud558\uace0 \uc5f0\uc18d\uc131\uc73c\ub85c\ubd80\ud130 \ud30c\uc0dd\ub418\ub294 \uc5ec\ub7ec \uac00\uc9c0 \uc131\uc9c8\ub4e4\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h3>\uc5f0\uc18d\ud568\uc218\uc758 \uc815\uc758<\/h3>\n<p>\uc5f0\uc18d\uc758 \uc815\uc758\ub294 \ud55c \uc810\uc5d0\uc11c\uc758 \uc5f0\uc18d\uacfc \uc9d1\ud569\uc5d0\uc11c\uc758 \uc5f0\uc18d\uc73c\ub85c \uad6c\ubd84\ud558\uc5ec \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uba3c\uc800 \ud55c \uc810\uc5d0\uc11c\uc758 \uc5f0\uc18d\uc758 \uc815\uc758\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1. (\uc810\uc5d0\uc11c\uc758 \uc5f0\uc18d\uc131; \uadf9\ud55c\uc744 \uc774\uc6a9\ud55c \uc815\uc758)<\/span><\/p>\n<ol class=\"bracket\">\n<li>\ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \uc774 \ub54c \ub9cc\uc57d<br \/>\n\\[\\lim_{x\\to c}f(x) = f(c)\\]<br \/>\n\uc774\uba74 \u2018\\(f\\)\ub294 <span class=\"defined\">\\(c\\)\uc5d0\uc11c \uc5f0\uc18d<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \uc67c\ucabd \ub05d\uc810\uc73c\ub85c \uac16\ub294 \ud55c \ubc18\ub2eb\ud78c \uad6c\uac04 \\( [ c , \\, d ) \\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d<br \/>\n\\[\\lim_{x\\rightarrow c^+}f(x) = f(c)\\]<br \/>\n\uc774\uba74 \u2018\\(f\\)\ub294 <span class=\"defined\">\\(c\\)\uc5d0\uc11c \uc6b0\uc5f0\uc18d<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \uc624\ub978\ucabd \ub05d\uc810\uc73c\ub85c \uac16\ub294 \ud55c \ubc18\uc5f4\ub9b0 \uad6c\uac04 \\((b,\\,c]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d<br \/>\n\\[\\lim_{x\\to c^-}f(x) = f(c)\\]<br \/>\n\uc774\uba74 \u2018\\(f\\)\ub294 <span class=\"defined\">\\(c\\)\uc5d0\uc11c \uc88c\uc5f0\uc18d<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\\(c\\)\uac00 \ud568\uc218 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc810\uc774\uace0 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2d0 \ub54c \u2018\\(f\\)\ub294 \\(c\\)\uc5d0\uc11c <span class=\"defined\">\ubd88\uc5f0\uc18d<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\ud568\uc218\uc758 \uc815\uc758\uc5ed\uc774 \ub2eb\ud78c \uad6c\uac04\uc778 \uacbd\uc6b0, \uad6c\uac04\uc758 \uc67c\ucabd \ub05d\uc810\uc5d0\uc11c\ub294 \ud568\uc218\uc758 \uc5f0\uc18d\uacfc \uc6b0\uc5f0\uc18d\uc744 \uac19\uc740 \uac1c\ub150\uc73c\ub85c \uac04\uc8fc\ud558\uba70, \uad6c\uac04\uc758 \uc624\ub978\ucabd \ub05d\uc810\uc5d0\uc11c\ub294 \ud568\uc218\uc758 \uc5f0\uc18d\uacfc \uc88c\uc5f0\uc18d\uc744 \uac19\uc740 \uac1c\ub150\uc73c\ub85c \uac04\uc8fc\ud55c\ub2e4.<\/p>\n<p>\ud568\uc218\uc758 \uc5f0\uc18d\uc131\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \\(\\epsilon &#8211; \\delta\\) \ub17c\ubc95\uc73c\ub85c \uc815\uc758\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<div class=\"definition\">\n<p>\n<span class=\"definition\">\uc815\uc758 2. (\uc810\uc5d0\uc11c\uc758 \uc5f0\uc18d\uc131; \\(\\epsilon-\\delta\\)\ub97c \uc774\uc6a9\ud55c \uc815\uc758)<\/span><\/p>\n<p>\ud568\uc218 \\(f : D \\to \\mathbb{R}\\)\uac00 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04, \ub610\ub294 \\(c\\)\ub97c \ub2eb\ud78c \ub05d\uc810\uc73c\ub85c \uac16\ub294 \ubc18\ub2eb\ud78c\uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\( |x-c| < \\delta\\)\uc778 \ubaa8\ub4e0 \\(x \\in D\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x) - f(c)| < \\epsilon\\)\uc774 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 <span class=\"defined\">\\(c\\)\uc5d0\uc11c \uc5f0\uc18d<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"remark\">\n<p>\n<span class=\"remark\">\ucc38\uace0.<\/span><br \/>\n\\(D\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc774\uace0 \\(c \\in D\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\((c-\\delta ,\\, c+\\delta) \\cap D\\)\uc758 \uc6d0\uc18c\uac00 \\(c\\) \ubc16\uc5d0 \uc5c6\uc744 \ub54c, \uc989 \\(c\\) \uc8fc\ubcc0\uc5d0 \\(D\\)\uc758 \uc6d0\uc18c\uac00 \uc5c6\uc744 \ub54c \\(c\\)\ub97c \\(D\\)\uc758 <span class=\"defined\">\uace0\ub9bd\uc810<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc815\uc758 1\uc5d0 \ub530\ub974\uba74 \\(c\\)\uac00 \ud568\uc218 \\(f : D \\to \\mathbb{R}\\)\uc758 \uc815\uc758\uc5ed\uc758 \uace0\ub9bd\uc810\uc77c \ub54c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ud55c\uc774 \uc815\uc758\ub418\uc9c0 \uc54a\uc73c\ubbc0\ub85c \\(c\\)\uc5d0\uc11c \\(f\\)\uc774 \uc5f0\uc18d\uc131 \ub610\ud55c \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ub098 \ub9cc\uc57d \uc815\uc758 2\uc5d0\uc11c \\(f\\)\uac00 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\ub294 \uac00\uc815\uc744 \uc81c\uac70\ud558\uba74 \\(c\\)\ub294 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uace0\ub9bd\uc810\uc774 \ub420 \uc218\ub3c4 \uc788\uc73c\uba70, \uc774 \uacbd\uc6b0 \uc815\uc758 2\uc5d0 \ub530\ub974\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774 \ub54c\ubb38\uc5d0 \ucc45\uc5d0 \ub530\ub77c\uc11c\ub294 \\(c\\)\uac00 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uace0\ub9bd\uc810\uc77c \ub54c \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \uc815\uc758\ud558\uae30\ub3c4 \ud55c\ub2e4.<br \/>\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<div class=\"definition\">\n<p>\n<span class=\"definition\">\uc815\uc758 3. (\uc9d1\ud569\uc5d0\uc11c\uc758 \uc5f0\uc18d\uc131)<\/span><\/p>\n<p>\ud568\uc218 \\(f : D \\to \\mathbb{R}\\)\uc640 \\(D\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(E\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(E\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f\\)\uac00 \uc5f0\uc18d\uc774\uba74 \u2018\\(f\\)\ub294 <span class=\"defined\">\\(E\\)\uc5d0\uc11c \uc5f0\uc18d<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub9cc\uc57d \\(f\\)\uac00 \uc815\uc758\uc5ed\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \u2018\\(f\\)\ub294 <span class=\"defined\">\uc5f0\uc18d\ud568\uc218<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><br \/>\n\\(p\\)\uac00 \ub2e4\ud56d\ud568\uc218\uc774\uba74 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(p\\)\uc758 \uadf9\ud55c\uac12\uc740 \ud568\uc22b\uac12\uacfc \uac19\uc73c\ubbc0\ub85c, \ubaa8\ub4e0 \ub2e4\ud56d\ud568\uc218\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \ubd84\uc218\ud568\uc218\ub294 \ubd84\ubaa8\uac00 \\(0\\)\uc774 \ub418\uc9c0 \uc54a\ub294 \uc810\uc5d0\uc11c \uadf9\ud55c\uac12\uacfc \ud568\uc22b\uac12\uc774 \uac19\uc73c\ubbc0\ub85c, \ubaa8\ub4e0 \ubd84\uc218\ud568\uc218\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4.<br \/>\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span><br \/>\n\ucd5c\ub300\uc815\uc218\ud568\uc218 \\(\\lfloor x \\rfloor\\)\ub294 \ubaa8\ub4e0 \uc815\uc22b\uc810\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\uba70, \uc815\uc218\uac00 \uc544\ub2cc \ubaa8\ub4e0 \uc810\uc5d0\uc11c\ub294 \uc5f0\uc18d\uc774\ub2e4. \uc815\uc22b\uc810\uc5d0\uc11c\ub294 \uc6b0\uc5f0\uc18d\uc774\uc9c0\ub9cc \uc88c\uc5f0\uc18d\uc740 \uc544\ub2c8\ub2e4. \ub610\ud55c \ubaa8\ub4e0 \uc815\uc22b\uc810\uc5d0\uc11c \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\uc774\ucc98\ub7fc \ud568\uc218 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\uc9c0\ub9cc \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc774 \ubaa8\ub450 \uc874\uc7ac\ud560 \ub54c \u2018\\(f\\)\ub294 \\(c\\)\uc5d0\uc11c <span class=\"defined\">\ub2e8\uc21c\ubd88\uc5f0\uc18d<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub2e8, \\(c\\)\uac00 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc67c\ucabd \ub05d\uc810\uc774\uac70\ub098 \uc624\ub978\ucabd \ub05d\uc810\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \ud55c\ubc29\ud5a5 \uadf9\ud55c\ub9cc \uc874\uc7ac\ud558\uc5ec\ub3c4 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \ub2e8\uc21c\ubd88\uc5f0\uc18d\uc774\ub2e4. \ub2e8\uc21c\ubd88\uc5f0\uc18d\uc774 \uc544\ub2cc \ubaa8\ub4e0 \ubd88\uc5f0\uc18d\uc744 <span class=\"defined\">\uc81c 2 \uc885 \ubd88\uc5f0\uc18d<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<br \/>\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<div class=\"remark\">\n<p><span class=\"remark\">\ucc38\uace0.<\/span><br \/>\n\ucc45\uc5d0 \ub530\ub77c\uc11c\ub294 \u2018\ub2eb\ud78c \uad6c\uac04\uc5d0\uc11c\uc758 \uc5f0\uc18d\uc131\u2019\uc744 \ub530\ub85c \uc815\uc758\ud558\uae30\ub3c4 \ud55c\ub2e4. \uc989 \u2018\ud568\uc218 \\(f\\)\uac00 \ub2eb\ud78c \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \uac83\u2019\uc744 \u2018\\(a < x < b\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0, \\(a\\)\uc5d0\uc11c \uc6b0\uc5f0\uc18d\uc774\uba70 \\(b\\)\uc5d0\uc11c \uc88c\uc5f0\uc18d\uc778 \uac83\u2019\uc73c\ub85c \uc815\uc758\ud558\uae30\ub3c4 \ud55c\ub2e4. \uc774\uc640 \uac19\uc774 \uc815\uc758\ud558\uba74 \\(a,\\) \\(b\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\uc9c0\ub9cc \\([a,\\,b]\\)\uc5d0\uc11c\ub294 \uc5f0\uc18d\uc778 \uacbd\uc6b0\uac00 \uc874\uc7ac\ud558\uac8c \ub41c\ub2e4. \uc608\ucee8\ub300 \ud568\uc218 \\(f:\\mathbb{R} \\to \\mathbb{R}\\)\ub97c\n\\[f(x) = \\begin{cases} 2 \\quad&#038;\\text{if} \\,\\, 1 \\le x \\le 3 \\\\[8pt]0\\quad&#038;\\text{otherwise}\\end{cases}\\]\n\uc73c\ub85c \uc815\uc758\ud558\uba74 \\(f\\)\ub294 \\(1,\\) \\(3\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\uc9c0\ub9cc \\([1,\\,3]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \ub41c\ub2e4.\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<h3>\uc5f0\uc18d\ud568\uc218\uc758 \uc131\uc9c8<\/h3>\n<p>\uadf9\ud55c\uc758 \ub300\uc218\uc801 \uc131\uc9c8\ub85c\ubd80\ud130 \ub2e4\uc74c\uacfc \uac19\uc740 \uc5f0\uc18d\ud568\uc218\uc758 \uc131\uc9c8\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p>\n<span class=\"theorem\">\uc815\ub9ac 1. (\uc5f0\uc18d\ud568\uc218\uc758 \ub300\uc218\uc801 \uc131\uc9c8)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \ubaa8\ub450 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<ul class=\"marginbottom0\">\n<li>\\(f+g ,\\) \\(f-g ,\\) \\( fg \\)\ub3c4 \ubaa8\ub450 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(k\\)\uac00 \uc2e4\uc218\uc774\uba74 \\(kf\\)\ub3c4 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(g(c) \\ne 0\\)\uc774\uba74 \\(f\/g\\)\ub3c4 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uba74 \\(f ^ n\\)\ub3c4 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<li>\\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uace0 \\(\\sqrt[n]{f}\\)\uac00 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0\uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uba74 \\(\\sqrt[n]{f}\\)\ub3c4 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/li>\n<\/ul>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(x\\,\\to\\,c\\)\uc77c \ub54c \uadf9\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n(f+g)(x) = (f(x)+g(x)) &#038;\\,\\to\\, (f(c)+g(c)) = (f+g)(c),\\\\[8pt]<br \/>\n(f-g)(x) = (f(x)-g(x)) &#038;\\,\\to\\, (f(c)-g(c)) = (f-g)(c),\\\\[8pt]<br \/>\n(fg)(x) = (f(x)g(x)) &#038;\\,\\to\\, (f(c)g(c)) = (fg)(c),\\\\[8pt]<br \/>\n(kf)(x) = (k f(x)) &#038;\\,\\to\\, (k f(c)) = (kf)(c),\\\\[8pt]<br \/>\n(f\/g)(x) = (f(x)\/g(x)) &#038;\\,\\to\\, (f(c)\/g(c)) = (f\/g)(c),\\\\[8pt]<br \/>\n(f^n )(x) = (f(x))^n &#038;\\,\\to\\, (f(c))^n = (f^n )(c),\\\\[8pt]<br \/>\n\\sqrt[n]{f}(x) = \\sqrt[n]{f(x)} &#038;\\,\\to\\, \\sqrt[n]{f(c)} = \\sqrt[n]{f}(c)<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c \uc815\ub9ac\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ud568\uc218\ub294 \ubaa8\ub450 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub450 \uc5f0\uc18d\ud568\uc218\ub97c \ud569\uc131\ud558\uc600\uc744 \ub54c \uadf8 \uacb0\uacfc\ub294 \uc5f0\uc18d\ud568\uc218\uac00 \ub41c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\ud569\uc131\ud568\uc218\uc758 \uc5f0\uc18d\uc131)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \ud568\uc218 \\(g\\)\uac00 \\(f(c)\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \ud569\uc131\ud568\uc218 \\(g \\circ f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(g\\)\uac00 \\(f(c)\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(\\delta_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\( |y-f(c)| < \\delta_1 \\)\uc778 \uc784\uc758\uc758 \\(y\\)\uc5d0 \ub300\ud558\uc5ec \\(|g(y) - g(f(c)) | < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\\(\\delta_1\\)\uc774 \uc591\uc218\uc774\uace0 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\( |x-c| < \\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x) - f(c)| < \\delta_1\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989\n\\[\\begin{align}\n|x-c| < \\delta \\quad \n&#038; \\Rightarrow \\quad |f(x) - f(c)| < \\delta_1 \\\\[8pt]\n&#038; \\Rightarrow \\quad |g(f(x)) - g(f(c))| < \\epsilon \\\\[8pt]\n&#038; \\Rightarrow \\quad |(g \\circ f)(x) - (g \\circ f)(c)| < \\epsilon\n\\end{align}\\]\n\uc774\ubbc0\ub85c \\(g \\circ f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\ud568\uc218\uc758 \uc5f0\uc18d\uc131\uacfc \ud569\uc131\ud568\uc218\uc758 \uadf9\ud55c)<\/span><\/p>\n<p>\ud568\uc218 \\(f,\\) \\(g\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(x \\,\\to\\, c\\)\uc77c \ub54c \\(f(x) \\,\\to\\, b\\)\uc774\uace0, \\(g\\)\uac00 \\(b\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\\[\\lim_{x\\,\\to\\,c} g(f(x)) = g(b) = g\\left( \\lim_{x\\,\\to\\,c} f(x) \\right) .\\]\n<\/p><\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(g\\)\uac00 \\(b\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(\\delta_1 > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\( |y-b| < \\delta_1 \\)\uc778 \uc784\uc758\uc758 \\(y\\)\uc5d0 \ub300\ud558\uc5ec \\(|g(y) - g(b) | < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\\(\\delta_1\\)\uc774 \uc591\uc218\uc774\uace0 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \\(b\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \\(\\delta > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\( 0<|x-c| < \\delta\\)\uc778 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x) - b| < \\delta_1\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989\n\\[\\begin{align}\n0<|x-c| < \\delta \\quad \n&#038; \\Rightarrow \\quad |f(x) - b| < \\delta_1 \\\\[8pt]\n&#038; \\Rightarrow \\quad |g(f(x)) - g(b)| < \\epsilon \\\\[8pt]\n&#038; \\Rightarrow \\quad |(g \\circ f)(x) - g (b)| < \\epsilon\n\\end{align}\\]\n\uc774\ubbc0\ub85c \\(g \\circ f\\)\ub294 \\(c\\)\uc5d0\uc11c \\(g(b)\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- \n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span>\n\n<span class=\"qee\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span>\n\n<span class=\"qee\"><\/span><\/p>\n\n<\/div>\n\n\n--><\/p>\n<h3>\uc5f0\uc18d\ud568\uc218\uc758 \uc0ac\uc787\uac12 \uc815\ub9ac<\/h3>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 4. (\uc5f0\uc18d\ud568\uc218\uc758 \uc0ac\uc787\uac12 \uc815\ub9ac)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(f(a) \\ne f(b)\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(C\\)\uac00 \\(f(a)\\)\uc640 \\(f(b)\\) \uc0ac\uc774\uc5d0 \uc788\ub294 \uac12\uc774\uba74 \\(f(c) = C\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(c\\)\uac00 \\((a,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(f(a) < C < f(b)\\)\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(f(a) > C > f(b)\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ud558\uba74 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uacfc \uac19\uc740 \uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[E = \\left\\{ x \\in [a,\\,b] \\,\\vert\\, f(x) \\le C \\right\\}\\]<br \/>\n\uc774 \uc9d1\ud569\uc740 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \uc2e4\uc218\uacc4\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \\(E\\)\uc758 \uc0c1\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8 \uc810\uc744 \\(c\\)\ub77c\uace0 \ud558\uc790. \uc774\uc81c \\(c\\)\uac00 \ubc14\ub85c \\(f(c)=C\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4\ub294 \uac83\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4.<\/p>\n<p>\\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c \\(1\/n\\)\uc740 \uc591\uc218\uc774\ubbc0\ub85c \uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[c- \\frac{1}{n} < x_n \\le c \\]\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(x_n\\)\uc774 \\(E\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(n\\,\\to\\,\\infty\\)\uc77c \ub54c \\(x_n\\,\\to\\,c\\)\uc774\uace0 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(n\\,\\to\\,\\infty\\)\uc77c \ub54c \\(f(x_n ) \\,\\to\\, f(c)\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_n \\in E\\) \uc989 \\(f(x_n ) \\le C\\)\uc774\ubbc0\ub85c \\(f(c) \\le C\\)\uc774\ub2e4.<\/p>\n<p>\ud55c\ud3b8 \\(c < b\\)\uc774\ubbc0\ub85c \\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c\n\\[c < y_n < c+ \\frac{1}{n}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(y_n\\)\uc774 \\([a,\\,b]\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(n\\,\\to\\,\\infty\\)\uc77c \ub54c \\(y_n\\,\\to\\,c\\)\uc774\uace0 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(n\\,\\to\\,\\infty\\)\uc77c \ub54c \\(f(y_n )\\,\\to\\,f(c)\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(y_n\\)\uc740 \\(E\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(C \\le f(y_n)\\) \uc989 \\(C \\le f(c)\\)\uc774\ub2e4.<\/p>\n<p>\uc694\ucee8\ub300 \\(f(c) \\le C \\le f(c)\\)\uc774\ubbc0\ub85c \\(f(c)=C\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<h3>\uc5f0\uc18d\ud568\uc218\uc758 \ucd5c\ub300 \ucd5c\uc18c \uc815\ub9ac<\/h3>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(M \\in E\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\le M\\)\uc774\uba74 \\(M\\)\uc744 \\(E\\)\uc758 \ucd5c\ub313\uac12\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \\(m \\in E\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(m\\le x\\)\uc774\uba74 \\(m\\)\uc744 \\(E\\)\uc758 \ucd5c\uc19f\uac12\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c \ud568\uc218\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 4. (\ud568\uc218\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12)<\/span><\/p>\n<p>\\(f\\)\uac00 \uc815\uc758\uc5ed\uc774 \\(D\\)\uc778 \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket\">\n<li>\ub9cc\uc57d \\(c_1 \\in D\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in D\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\le f(c_1 )\\)\uc774\uba74 \u2018\\(f\\)\ub294 \\(c_1\\)\uc5d0\uc11c <span class=\"defined\">\ucd5c\ub313\uac12<\/span>\uc744 \uac00\uc9c4\ub2e4\u2019 \ub610\ub294 \u2018\\(f\\)\ub294 \\(D\\) \uc704\uc5d0\uc11c <span class=\"defined\">\ucd5c\ub313\uac12<\/span> \\(f(c_1)\\)\uc744 \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(c_2 \\in D\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in D\\)\uc5d0 \ub300\ud558\uc5ec \\(f(c_2 ) \\le f(x )\\)\uc774\uba74 \u2018\\(f\\)\ub294 \\(c_2\\)\uc5d0\uc11c <span class=\"defined\">\ucd5c\uc19f\uac12<\/span>\uc744 \uac00\uc9c4\ub2e4\u2019 \ub610\ub294 \u2018\\(f\\)\ub294 \\(D\\) \uc704\uc5d0\uc11c <span class=\"defined\">\ucd5c\uc19f\uac12<\/span> \\(f(c_2)\\)\ub97c \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\ub2eb\ud78c \uad6c\uac04\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218\ub294 \uadf8 \uad6c\uac04\uc5d0\uc11c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 5. (\uc5f0\uc18d\ud568\uc218\uc758 \ucd5c\ub300 \ucd5c\uc18c \uc815\ub9ac)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\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>\uba3c\uc800 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc784\uc744 \ubcf4\uc774\uc790. \uc989 \uc591\uc218 \\(X\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x\\in[a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert f(x) \\rvert\\le X\\)\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc790.<\/p>\n<p>\uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x_n ) > n\\)\uc778 \uc810 \\(x_n\\)\uc774 \\([a,\\,b]\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uc774 \ub54c \\(\\left\\{x_n \\right\\}\\)\uc740 \ubaa8\ub4e0 \ud56d\uc774 \\([a,\\,b]\\)\uc5d0 \uc18d\ud558\ub294 \uc218\uc5f4\uc774\ubbc0\ub85c \uc720\uacc4\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc720\uacc4 \uc218\uc5f4\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{x_n \\right\\}\\)\uc758 \ud56d\uc744 \uc77c\ubd80\ub97c \ubaa8\uc544\uc11c \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc744 \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uadf8 \uc218\uc5f4\uc744 \\(\\left\\{y_k \\right\\}\\)\ub77c\uace0 \ud558\uc790. \\([a,\\,b]\\)\uac00 \ub2eb\ud78c \uad6c\uac04\uc774\ubbc0\ub85c \\(\\left\\{y_k \\right\\}\\)\uc758 \uadf9\ud55c\uac12\uc740 \\([a,\\,b]\\)\uc5d0 \uc18d\ud55c\ub2e4. \uadf8 \uadf9\ud55c\uac12\uc744 \\(c\\)\ub77c\uace0 \ud558\uc790. \\(\\left\\{y_k \\right\\}\\)\ub294 \\(f(x_n ) > n\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc218\uc5f4 \\(\\left\\{x_n \\right\\}\\)\uc758 \uc77c\ubd80 \ud56d\uc744 \ubaa8\uc544 \ub9cc\ub4e0 \uc218\uc5f4\uc774\ubbc0\ub85c \\(k\\,\\to\\,\\infty\\)\uc77c \ub54c \\(f(y_k )\\,\\to\\,\\infty\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(f\\)\ub294 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(k\\,\\to\\,\\infty\\)\uc77c \ub54c \\(y_k \\,\\to\\,c\\)\uc774\ubbc0\ub85c \\(f(y_k ) \\,\\to\\, f(c)\\)\uc774\ub2e4. \uc774\uac83\uc740 \\(f(y_k )\\,\\to\\,\\infty\\)\ub77c\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\uc774\ub2e4.<\/p>\n<p>\uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc784\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774\uc81c \ub2e4\uc74c\uacfc \uac19\uc740 \uc9d1\ud569\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[E = \\left\\{ f(x) \\,\\vert\\, x \\in [a,\\,b]\\right\\}\\]<br \/>\n\uadf8\ub7ec\uba74 \\(E\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \uc2e4\uc218\uacc4\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \\(E\\)\uc758 \uc0c1\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8 \uc0c1\ud55c\uc744 \\(M\\)\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(M\\)\uc774 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ucd5c\ub313\uac12\uc774 \uc544\ub2c8\ub77c\uba74, \uc784\uc758\uc758 \\(x\\in[a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) < M\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(x\\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec\n\\[h(x) = \\frac{1}{M - f(x)}\\]\n\uc774\ub77c\uace0 \ud558\uba74 \\(h\\)\ub294 \\([a,\\,b]\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc798 \uc815\uc758\ub41c \uc5f0\uc18d\ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc55e\uc758 \ub17c\uc758\uc5d0 \uc758\ud558\uc5ec \\(h\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \uc989 \uc591\uc218 \\(Y\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x\\in[a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(h(x) < Y\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\uac83\uc740\n\\[\\frac{1}{M-f(x)} < Y\\]\n\uc989\n\\[f(x) < M - \\frac{1}{Y}\\]\n\uc774 \uc131\ub9bd\ud568\uc744 \uc758\ubbf8\ud55c\ub2e4. \ud55c\ud3b8 \\(M\\)\uc740 \\(E\\)\uc758 \uc0c1\ud55c\uc774\uace0, \\(M - 1\/Y\\)\ub294 \\(M\\)\ubcf4\ub2e4 \ub354 \uc791\uc740 \uac12\uc740 \uac12\uc774\ubbc0\ub85c \\(f(x) > M-1\/y\\)\uc778 \uc810 \\(x\\)\uac00 \\([a,\\,b]\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(M\\)\uc740 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ucd5c\ub313\uac12\uc774\ub2e4.<\/p>\n<p>\uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ucd5c\uc19f\uac12\uc774 \uc874\uc7ac\ud568\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.<br \/>\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<h3>\uc5ed\ud568\uc218\uc758 \uc5f0\uc18d\uc131<\/h3>\n<p>\ud568\uc218\uac00 \uc5f0\uc18d\uc774\ub77c\ub294 \uac83\uc740 \uc9c1\uad00\uc801\uc73c\ub85c \uadf8 \uadf8\ub798\ud504\uac00 \ub04a\uc5b4\uc9c0\uc9c0 \uc54a\uace0 \uc774\uc5b4\uc838 \uc788\ub2e4\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud568\uc218\uac00 \uc5f0\uc18d\uc774\uace0 \uc5ed\ud568\uc218\uac00 \uc874\uc7ac\ud560 \uadf8 \uc5ed\ud568\uc218\uc758 \uadf8\ub798\ud504 \ub610\ud55c \ub04a\uc5b4\uc9c0\uc9c0 \uc54a\uace0 \uc774\uc5b4\uc838 \uc788\uac8c \ub41c\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p>\n<span class=\"theorem\">\uc815\ub9ac 6. (\uc5ed\ud568\uc218\uc758 \uc5f0\uc18d\uc131)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \ub2eb\ud78c \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc77c\ub300\uc77c\uc778 \ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(f\\)\uc758 \uce58\uc5ed\uc744 \\(J\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc5ed\ud568\uc218 \\(f^{-1} :J \\to I\\) \ub610\ud55c \uc5f0\uc18d\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc774 \uc815\ub9ac\ub97c \uc99d\uba85\ud558\uae30 \uc704\ud574\uc11c\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ubcf4\uc870\uc815\ub9ac\uac00 \ud544\uc694\ud558\ub2e4.<\/p>\n<div class=\"theorem\">\n<p>\n<span class=\"theorem\">\uc815\ub9ac 7. (\ub2e8\uc870\uc218\ub834 \uc815\ub9ac)<\/span><\/p>\n<p>\\(h\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc99d\uac00\ud568\uc218\uc774\uace0 \\(d\\)\uac00 \\(I\\)\uc758 \ub05d\uc810\uc740 \uc544\ub2cc \uc810\uc774\ub77c\uba74 \\(d\\)\uc5d0\uc11c \\(h\\)\uc758 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc740 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4. \ub9cc\uc57d \\(d\\)\uac00 \\(I\\)\uc758 \uc67c\ucabd \ub05d\uc810\uc774\uace0 \\(h\\)\uac00 \\(I\\)\uc5d0\uc11c \uc544\ub798\ub85c \uc720\uacc4\ub77c\uba74 \\(d\\)\uc5d0\uc11c \\(I\\)\uc758 \uc6b0\uadf9\ud55c\uc774 \uc218\ub834\ud558\uba70, \ub9cc\uc57d \\(d\\)\uac00 \\(I\\)\uc758 \uc624\ub978\ucabd \ub05d\uc810\uc774\uace0 \\(h\\)\uac00 \\(I\\)\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\ub77c\uba74 \\(d\\)\uc5d0\uc11c \\(I\\)\uc758 \uc88c\uadf9\ud55c\uc774 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(d\\)\uac00 \\(I\\)\uc758 \ub05d\uc810\uc774 \uc544\ub2cc \uc810\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc9d1\ud569 \\[E = \\left\\{ h(x) \\,\\vert\\, x \\in I ,\\, x < d \\right\\}\\]\ub97c \uc0dd\uac01\ud558\uc790. \uc774 \uc9d1\ud569\uc740 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \uc2e4\uc218\uacc4\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \\(E\\)\uc758 \uc0c1\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(E\\)\uc758 \uc0c1\ud55c\uc740 \\(E\\)\uc758 \uc6d0\uc18c\uc77c \uc218\ub3c4 \uc788\uace0 \uadf8\ub807\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\uc9c0\ub9cc, \uadf8\uac83\uc740 \uc0c1\uad00 \uc5c6\ub2e4. \uadf8 \uc810\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(L\\)\uc774 \\(E\\)\uc758 \uc0c1\ud55c\uc774\ubbc0\ub85c \\(L &#8211; \\epsilon < h(a) \\le L,\\) \\(a < d\\)\uc778 \uc810 \\(a\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(\\delta = d-a\\)\ub77c\uace0 \ud558\uba74 \\(d - \\delta < x < d\\)\uc77c \ub54c\n\\[L - \\epsilon < h(x) \\le L\\]\n\uc774\ubbc0\ub85c \\(d\\)\uc5d0\uc11c \\(h\\)\uc758 \uc88c\uadf9\ud55c\uc740 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(d\\)\uc5d0\uc11c \\(h\\)\uc758 \uc6b0\uadf9\ud55c\uc774 \uc874\uc7ac\ud568\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.<\/p>\n<p>\\(d\\)\uac00 \\(I\\)\uc758 \ub05d\uc810\uc778 \uacbd\uc6b0\uc758 \uc99d\uba85\ub3c4 \uc774\uc640 \ube44\uc2b7\ud558\ub2e4. \uc99d\uba85\uc744 \uc5ec\uae30\uc5d0 \ub108\ubb34 \uc0c1\uc138\ud788 \uc368\uc8fc\uba74 \uc7ac\ubbf8\uac00 \uc5c6\uc744\ud14c\ub2c8 \uc9c1\uc811 \uc99d\uba85\ud574\ubcf4\uae30 \ubc14\ub780\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc774\uc81c \uc5ed\ud568\uc218\uc758 \uc5f0\uc18d\uc131\uc744 \uc99d\uba85\ud558\uc790.<\/p>\n<div class=\"proof\">\n<p style=\"padding-left: 0;\"><span class=\"proof\">\uc815\ub9ac 6\uc758 \uc99d\uba85.<\/span> \ub450 \ub2e8\uacc4\ub85c \uc99d\uba85\ud55c\ub2e4.<\/p>\n<p><span class=\"proof\">1\ub2e8\uacc4.<\/span><br \/>\n\\(f\\)\uac00 \ub2eb\ud78c \uad6c\uac04\uc5d0\uc11c \uc77c\ub300\uc77c \ud568\uc218\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc774\uac70\ub098 \ub610\ub294 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uac10\uc18c\ud568\uc218\uc774\ub2e4. \uc774 \uc0ac\uc2e4\uc744 \uc99d\uba85\ud558\uc790. \uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\ub3c4 \uc544\ub2c8\uace0 \uac10\uc18c\ud568\uc218\ub3c4 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(I\\)\uc758 \uc138 \uc810 \\(x_1 ,\\) \\(x_2 ,\\) \\(x_3 \\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(x_1 < x_2 < x_3\\)\uc774\uba74\uc11c \\(f(x_1 ) < f(x_2 ) > f(x_3 )\\)\uc774\uac70\ub098, \\(x_1 < x_2 < x_3\\)\uc774\uba74\uc11c \\(f(x_1 ) > f(x_2 ) < f(x_3 )\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub450 \uac00\uc9c0 \uc911 \uccab \ubc88\uc9f8 \uacbd\uc6b0\uc5d0 \ub300\ud558\uc5ec \uc99d\uba85\ud558\uc790. \\(f(x_1 ) < f(x_2 ) > f(x_3 )\\)\uc778 \uacbd\uc6b0,<br \/>\n\\[C = \\frac{f(x_2 ) + \\max \\left\\{ f(x_1 ) ,\\, f(x_3 ) \\right\\}}{2}\\]<br \/>\n\ub77c\uace0 \ud558\uba74<br \/>\n\\[f(x_1 ) < C < f(x_2 ) ,\\qquad f(x_2 ) > C > f(x_3 )\\]<br \/>\n\uc774\ubbc0\ub85c \uc5f0\uc18d\ud568\uc218\uc758 \uc0ac\uc787\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[ x_1 < p_1 < x_2 < p_2 < x_3 \\]\n\uc778 \ub450 \uc810 \\(p_1 ,\\) \\(p_2 \\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[ f(p_1 ) = C = f(p_2 )\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774\uac83\uc740 \\(f\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\ub77c\ub294 \ub370\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \\(f(x_1 ) > f(x_2 ) < f(x_3 )\\)\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \ubaa8\uc21c\uc744 \ub04c\uc5b4\ub0bc \uc218 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc774\uac70\ub098 \uac10\uc18c\ud568\uc218\uc77c \uc218\ubc16\uc5d0 \uc5c6\ub2e4.<\/p>\n<p><span class=\"proof\">2\ub2e8\uacc4.<\/span><br \/>\n\uc774\uc81c \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f^{-1}\\)\ub294 \\(J\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc774\ub2e4.<br \/>\n\\(d \\in J\\)\uc774\uace0 \\(d\\)\uac00 \\(J\\)\uc758 \ub05d\uc810\uc740 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(f^{-1}\\)\uac00 \\(d\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \\(f^{-1}\\)\ub294 \uc99d\uac00\ud568\uc218\uc774\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(d\\)\uc5d0\uc11c \\(f^{-1}\\)\uc758 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uba70, \ub458 \uc911 \ud558\ub098\ub294 \\(f^{-1}(d)\\)\uc640 \ub2e4\ub974\ub2e4. \\(d\\)\uc5d0\uc11c \\(f^{-1}\\)\uc758 \uc88c\uadf9\ud55c\uc774 \\(p\\)\uc774\uace0 \uadf8 \uac12\uc774 \\(f^{-1}(d)\\)\uc640 \ub2e4\ub974\ub2e4\uace0 \ud558\uc790. \\(c = f^{-1}(d)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(p < c\\)\uc774\uba70, \\(p < x < c\\)\uc778 \\(x\\)\ub294 \\(I\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\uac8c \ub41c\ub2e4. \uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \\(p < x < c\\)\uc778 \\(x\\)\uac00 \\(I\\)\uc5d0 \uc18d\ud55c\ub2e4\uba74 \\(y=f(x)\\)\uc5d0 \ub300\ud558\uc5ec\n\\[ y < d \\quad \\text{and} \\quad f^{-1}(y) < c = f^{-1}(d)\\]\n\uc774\ubbc0\ub85c \\(d\\)\uc5d0\uc11c \\(f^{-1}\\)\uc758 \uc88c\uadf9\ud55c\uc740 \\(f^{-1}(y)\\) \uc774\uc0c1\uc774 \ub418\uc5b4\uc57c \ud55c\ub2e4. \uadf8\ub7f0\ub370 \uc774\uac83\uc740 \\[f^{-1} (d^- ) = p < f^{-1}(y)\\]\uc774\ubbc0\ub85c \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(d\\)\uc5d0\uc11c \\(f^{-1}\\)\uc758 \uc88c\uadf9\ud55c\uc740 \\(f^{-1}(d)\\)\uc640 \ub2e4\ub97c \uc218 \uc5c6\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(d\\)\uc5d0\uc11c \\(f^{-1}\\)\uc758 \uc6b0\uadf9\ud55c\ub3c4 \\(f^{-1}(d)\\)\uc640 \ub2e4\ub97c \uc218 \uc5c6\ub2e4. \uc774\uac83\uc740 \\(f^{-1}\\)\uac00 \\(d\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4. \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uac10\uc18c\ud568\uc218\uc778 \uacbd\uc6b0\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(f^{-1}\\)\uac00 \\(d\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc774 \uc99d\uba85\ub41c\ub2e4.<\/p>\n<p>\\(d\\)\uac00 \\(J\\)\uc758 \uc67c\ucabd \ub05d\uc810\uc774\uac70\ub098 \uc624\ub978\ucabd \ub05d\uc810\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uc704\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(d\\)\uc5d0\uc11c \\(f^{-1}\\)\uc758 \uadf9\ud55c\uac12\uacfc \ud568\uc22b\uac12\uc774 \uac19\uc74c\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.<br \/>\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc9c1\uad00\uc801\uc73c\ub85c, \ud568\uc218\uac00 \uc5f0\uc18d\uc774\ub77c\ub294 \uac83\uc740 \uadf8 \uadf8\ub798\ud504\uac00 \ub04a\uc5b4\uc9c0\uc9c0 \uc54a\uace0 \uc774\uc5b4\uc838 \uc788\ub294 \uac83\uc774\ub2e4. \uadf8\ub7ec\ub098 \uc774\uc640 \uac19\uc740 \uc9c1\uad00\uc801 \uac1c\ub150\ub9cc\uc73c\ub85c\ub294 \uba85\ud655\ud558\uac8c \ub2e4\ub8f0 \uc218 \uc5c6\ub294 \uc5f0\uc18d\ud568\uc218\uc758 \uc131\uc9c8\ub4e4\uc774 \uc788\ub2e4. \uc774 \uae00\uc5d0\uc11c\ub294 \uc5f0\uc18d\uc131\uc744 \uc5c4\ubc00\ud558\uac8c \uc815\uc758\ud558\uace0 \uc5f0\uc18d\uc131\uc73c\ub85c\ubd80\ud130 \ud30c\uc0dd\ub418\ub294 \uc5ec\ub7ec \uac00\uc9c0 \uc131\uc9c8\ub4e4\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc5f0\uc18d\ud568\uc218\uc758 \uc815\uc758 \uc5f0\uc18d\uc758 \uc815\uc758\ub294 \ud55c \uc810\uc5d0\uc11c\uc758 \uc5f0\uc18d\uacfc \uc9d1\ud569\uc5d0\uc11c\uc758 \uc5f0\uc18d\uc73c\ub85c \uad6c\ubd84\ud558\uc5ec \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uba3c\uc800 \ud55c \uc810\uc5d0\uc11c\uc758 \uc5f0\uc18d\uc758 \uc815\uc758\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uc815\uc758 1. (\uc810\uc5d0\uc11c\uc758 \uc5f0\uc18d\uc131; \uadf9\ud55c\uc744 \uc774\uc6a9\ud55c \uc815\uc758) \ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \uc6d0\uc18c\ub85c&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[153,154,160,157,162,133,158,132,152,156,161,151,155,159],"class_list":["post-1934","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-continuity","tag-discontinuity","tag-extreme-value-theorem","tag-intermediate-value-theorem","tag-inverse-function-theorem","tag-monotone-convergence","tag-158","tag-132","tag-152","tag-156","tag-161","tag-151","tag-155","tag-159"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1934","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=1934"}],"version-history":[{"count":73,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1934\/revisions"}],"predecessor-version":[{"id":7897,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1934\/revisions\/7897"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1934"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1934"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1934"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}