{"id":168,"date":"2019-08-22T20:34:48","date_gmt":"2019-08-22T20:34:48","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=168"},"modified":"2020-06-17T18:56:06","modified_gmt":"2020-06-17T18:56:06","slug":"continuity","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/continuity\/","title":{"rendered":"Continuity"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Continuity &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nFor functions that are &#8220;normal&#8221; enough, we know immediately whether\nor not they are continuous at a given point.  Nevertheless, the\ncontinuity of a function is such an important property that we need a\nprecise definition of continuity at a point:\n<\/p>\n\n\n\n<center>\n\tA function $f$ is <b>continuous at $c$<\/b> if and only if \n\t$\\displaystyle\\lim_{x\\to c} f(x)=f(c).$\n<\/center>\n\n\n\n<p>\nThat is, $f$ is continuous at $c$ if and only if for all $\\varepsilon &gt; 0$\nthere exists a $\\delta &gt; 0$ such that \n\\[{\\small\\textrm{if }} |x-c|&lt;\\delta \\quad{\\small\\textrm{then }}\n|f(x)-f(c)|&lt;\\varepsilon.\\]\nIn words, for $x$ close to $c$, $f(x)$ should be close to $f(c)$.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Notes<\/h4>\n\n\n\n<ul class=\"wp-block-list\"><li> If $f$ is continuous at every real number $c$, then $f$ is said\n\t\tto be <b>continuous<\/b>.\n\t<\/li><li> If $f$ is <i>not<\/i> continuous at $c$, then $f$ is said to be\n\t\t<b>discontinuous at $c$<\/b>.  The function $f$ can be discontinuous\n\t\tfor two distinct reasons:\n\t\t<ul>\n\t\t\t<li> $f(x)$ does not have a limit as $x\\to c$.  (Specifically, if the\n\t\t\t\tleft- and right-hand limits exist but are different, the discontinuity\n\t\t\t\tis called a <b>jump discontinuity<\/b>.)\n\t\t\t<\/li><li> $f(x)$ has a limit as $x\\to c$, but $\\lim_{x\\to c} f(x)\\neq\n\t\t\t\tf(c)$ or $f(c)$ is undefined.  (This is called a <b>removable\n\t\t\t\tdiscontinuity<\/b>, since we can &#8220;remove&#8221; the discontinuity at $c$ by\n\t\t\t\tredefining $f(c)$ as $\\lim_{x\\to c} f(x)$.)\n\t\t<\/li><\/ul>\n<\/li><\/ul>\n\n\n<style>.kt-accordion-id_3c5b73-66 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_3c5b73-66 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_3c5b73-66 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_3c5b73-66 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_3c5b73-66 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_3c5b73-66 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_3c5b73-66 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3c5b73-66:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_3c5b73-66 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_3c5b73-66 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_3c5b73-66 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_e715ca-a5\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Figures<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-media-text alignwide\" style=\"grid-template-columns:35% auto\"><figure class=\"wp-block-media-text__media\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"109\" height=\"121\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/Cont_figure1.gif?resize=109%2C121&#038;ssl=1\" alt=\"Discontinuity at x=0\" class=\"wp-image-339\"\/><\/figure><div class=\"wp-block-media-text__content\">\n<p class=\"has-medium-font-size\">$f(x) = \\frac{1}{x}$<br>Disconinuity at $x = 0$<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-media-text alignwide\" style=\"grid-template-columns:35% auto\"><figure class=\"wp-block-media-text__media\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"109\" height=\"121\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/Cont_figure2.gif?resize=109%2C121&#038;ssl=1\" alt=\"Removable Discontinuity at x=3\" class=\"wp-image-340\"\/><\/figure><div class=\"wp-block-media-text__content\">\n<p class=\"has-medium-font-size\">$f(x) = \\frac{x^2-9}{x-3}$<br>Removable Discontinuity at $x = 3$<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-media-text alignwide\" style=\"grid-template-columns:35% auto\"><figure class=\"wp-block-media-text__media\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"109\" height=\"121\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/Cont_figure3.gif?resize=109%2C121&#038;ssl=1\" alt=\"Discontinuity at x=0\" class=\"wp-image-341\"\/><\/figure><div class=\"wp-block-media-text__content\">\n<p class=\"has-medium-font-size\"> \\[ f(x)=\\left\\{\\begin{array}{ll} x^2, &amp; x &lt; 0\\\\ 1, &amp; x = 0\\\\x + 1, &amp; x &gt;  0\\end{array}\\right. \\]  <br>Discontinuity at $x = 0$<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-media-text alignwide\" style=\"grid-template-columns:35% auto\"><figure class=\"wp-block-media-text__media\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"109\" height=\"124\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/Cont_figure4.gif?resize=109%2C124&#038;ssl=1\" alt=\"Discontinuity at x=0\" class=\"wp-image-342\"\/><\/figure><div class=\"wp-block-media-text__content\">\n<p class=\"has-medium-font-size\">\\[ f(x)=\\left\\{\\begin{array}{ll} 1, &amp; x &lt; 0\\\\ 3, &amp; x \\leq 0\\end{array}\\right. \\]  <br>Jump Discontinuity at $x = 0$ <\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-media-text alignwide\" style=\"grid-template-columns:35% auto\"><figure class=\"wp-block-media-text__media\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"109\" height=\"119\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/Cont_figure5.gif?resize=109%2C119&#038;ssl=1\" alt=\"Discontinuity at x=0\" class=\"wp-image-343\"\/><\/figure><div class=\"wp-block-media-text__content\">\n<p class=\"has-medium-font-size\">$f(x) = sin(\\frac{1}{x})$<br>Discontinuity at $x = 0$<\/p>\n<\/div><\/div>\n\n\n\n<p><\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p>\n\nRather than returning to the $\\varepsilon$-$\\delta$ definition\nwhenever we want to prove a function is continuous at a point, we\nbuild up our collection of continuous functions by combining functions\nwe know are continuous:\n\n<\/p>\n\n\n\n<p>\nIf $f$ and $g$ are continuous at $c$, then\n<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li> $f+g$ is continuous at $c$.\n\t<\/li><li> $\\alpha f$ is continuous at $c$ for any real number $\\alpha$.\n\t<\/li><li> $fg$ is continous at $c$.\n\t<\/li><li> $f\/g$ is continuous at $c$ if $g(c)\\neq 0$.\n<\/li><\/ol>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nThe function $\\displaystyle f(x)=\\frac{x^2-4}{(x-2)(x-1)}$ is\ncontinuous everywhere except at $x=2$ and at $x=1$.  The discontinuity\nat $x=2$ is removable, since $\\displaystyle \\frac{x^2-4}{(x-2)(x-1)}$\ncan be simplified to $\\displaystyle \\frac{x+2}{x-1}$.  To remove the\ndiscontinuity, define \n\\[f(2)=\\frac{2+2}{2-1}=4.\\]\n\n<\/p>\n\n\n\n<p>\nWe can also look at the composition $f\\circ g$ of two functions,\n\\[(f\\circ g)(x)=f(g(x)).\\]\n\n<\/p>\n\n\n\n<p> If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$, then the composition $f\\circ g$ is continuous at $c$. <\/p>\n\n\n<style>.kt-accordion-id_16fbbf-ae .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_16fbbf-ae .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_16fbbf-ae > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_16fbbf-ae > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_16fbbf-ae .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_16fbbf-ae .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_16fbbf-ae > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_16fbbf-ae:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_16fbbf-ae .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_16fbbf-ae .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_16fbbf-ae kt-accordion-has-3-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-3 kt-pane_b3d736-25\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Proof<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n<title>Theorem<\/title>\n\n\n\n<h4 class=\"wp-block-heading\">Theorem<\/h4>\n\n\n\n<p>If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$, then the composition $f\\circ g$ is continuous at $c$.  <\/p>\n\n\n\n<h5 class=\"wp-block-heading\">Idea of Proof:<\/h5>\n\n\n\n<p> For $x$ &#8220;close&#8221; to $c$, $g(x)$ is &#8220;close&#8221; to $g(c)$.<br> For $g(x)$ &#8220;close&#8221; to $g(c)$, $f(g(x))$ is &#8220;close&#8221; to $f(g(c))$.<br> So for $x$ &#8220;close&#8221; to $c$, $f(g(x))$ is &#8220;close&#8221; to $f(g(c))$.  <\/p>\n\n\n\n<h5 class=\"wp-block-heading\">Proof:<\/h5>\n\n\n\n<p>\n\nLet $\\varepsilon&gt;0$.\n\n<\/p>\n\n\n\n<p>We will show that there exists $\\delta&gt;0$ such that if $|x-c|&lt;\\delta$, then $|f(g(x))-f(g(c))|&lt;\\varepsilon$.<\/p>\n\n\n\n<p>\nSince $f$ is continuous at $g(c)$, there exists $\\delta_1&gt;0$ such that\n<\/p>\n\n\n\n<p>\n\t$$ {\\small\\textrm{if }} |x_1-g(c)|&lt;\\delta_1, {\\small\\textrm{ then }} |f(x_1)-f(g(c))|&lt;\\varepsilon. $$\n\nFor $\\delta_1&gt;0$, there exists $\\delta&gt;0$ such that \n<\/p>\n\n\n\n<p>\n\t$$ {\\small\\textrm{if }} |x-c|&lt;\\delta, {\\small\\textrm{ then }} |g(x)-g(c)|&lt;\\delta_1 $$\n\nby the continuity of $g$ at $c$.\n\n<\/p>\n\n\n\n<p>\nSubstituting $g(x)$ for $x_1$ in (1), we have from (2)\nand (1) that\n\\[{\\small\\textrm{if }} |x-c|&lt;\\delta, {\\small\\textrm{ then }} |f(g(x))-f(g(c))|&lt;\\varepsilon,\\]\nand the proof is complete.\n\n<\/p>\n\n\n\n<p>\n(This proof is taken from Salas and Hille&#8217;s <i>Calculus: One\nVariable<\/i>, 7th ed.)\n\n\n<!------------------------>\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p>\n\nWe&#8217;d also like to speak of continuity on a closed interval $[a,b]$.\nTo deal with the endpoints $a$ and $b$, we define \n<b>one-sided continuity<\/b>:\n\n<\/p>\n\n\n\n<p>\nA function $f$ is <b>continuous from the left at $c$<\/b> if and only\nif $\\displaystyle\\lim_{x\\to c^-} f(x)=f(c)$.  It is <b>continuous from the\nright at $c$<\/b> if and only if $\\displaystyle\\lim_{x\\to c^+} f(x)=f(c)$.\n\n<\/p>\n\n\n\n<p>\nWe say that $f$ is <b>continuous on $[a,b]$<\/b> if and only if\n<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li> $f$ is continuous on $(a,b)$, <\/li><li> $f$ is continuous from the right at $a$, and <\/li><li> $f$ is continuous from the left at $b$.<\/li><\/ol>\n\n\n<style>.kt-accordion-id_924682-04 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_924682-04 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_924682-04 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap 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.kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_924682-04 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_924682-04 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_924682-04:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_924682-04 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_924682-04 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_924682-04 kt-accordion-has-3-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-3 kt-pane_d558e5-bc\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Figure 2<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n<title>One-Sided Continuity<\/title>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"162\" height=\"157\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/Cont_figure6.gif?resize=162%2C157&#038;ssl=1\" alt=\"A graph of f(x) as x approaches infinity from 0\" class=\"wp-image-344\"\/><\/figure><\/div>\n\n\n\n<p>  $\\displaystyle{\\lim_{x\\rightarrow 0^+} f(x)=f(0)}$ so $f(x) = \\sqrt{x}$ is <em>continuous from the right at $0$<\/em>.  <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"162\" height=\"157\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/Cont_figure7.gif?resize=162%2C157&#038;ssl=1\" alt=\"A graph of f(x) as x approaches -1 from negative infinity\" class=\"wp-image-345\"\/><\/figure><\/div>\n\n\n\n<p>$\\displaystyle{\\lim_{x\\rightarrow 1^-} f(x)=f(1)}$ so $f(x)=\\sqrt{1-x}$ is <em>continuous from the left at $1$<\/em>.  <\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p>\n\nNote that $f$ is continuous at $c$ if and only if the right- and\nleft-hand limits exist and both equal $f(c)$.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"273\" height=\"231\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/Cont_figure8.gif?resize=273%2C231&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-346\"\/><\/figure><\/div>\n\n\n\n<p> The function \\[ f(x)=\\left\\{\\begin{array}{ll} x, &amp; x\\leq 0\\\\ x^2, &amp; 0 &lt; x\\leq 1\\\\ \\frac{2}{x}, &amp; 1 &lt; x\\leq 2\\\\ x-1, &amp; x &gt; 2 \\end{array}\\right. \\] is continuous everywhere except at $x=1$, where $f$ has a jump discontinuity. <br> <\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<h4>Key Concepts<\/h4>\n<\/center>\n\n\n\n<p>\n\nA function $f$ is continuous at $c$ if and only if \n$\\displaystyle\\lim_{x \\to c} f(x) = f(c)$. \n\n<\/p>\n\n\n\n<p>\nThat is, $f$ is <b>continuous<\/b> at $c$ if and only if for all \n$\\varepsilon &gt; 0$ there exists a $\\delta &gt; 0$ such that\n\n\\[{\\small\\textrm{if }} |x-c| &lt; \\delta {\\small\\textrm{  then }} |f(x)-f(c)| &lt; \n\\varepsilon{\\small\\textrm{.}}\\]\n\nIn words, for $x$ close to $c$, $f(x)$ should be close to $f(c)$.\n\n\n\n<!------------------------>\n\n\n<br>\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ1010\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Continuity &#8211; HMC Calculus Tutorial For functions that are &#8220;normal&#8221; enough, we know immediately whether or not they are continuous at a given point. Nevertheless, the continuity of a function is such an important property that we need a precise definition of continuity at a point: A function $f$ is continuous at $c$ if and&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-168","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/168","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=168"}],"version-history":[{"count":9,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/168\/revisions"}],"predecessor-version":[{"id":1210,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/168\/revisions\/1210"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=168"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}