Answer:<\/strong><\/p>\n\n\n\n A function that does not satisfy continuity<\/em> is:1 & \\text{if } x < 0 \\\\ 2 & \\text{if } x \\geq 0 \\end{cases} \\] A function that does not satisfy *smoothness* (but is still continuous) is: \\[ f(x) = |x| \\] — **Explanation (approx. 300 words):** Differentiability of a function at a point requires two essential conditions to be met: the function must be *continuous* at that point, and it must be *smooth*, meaning that there is no sharp turn or cusp in the graph. Both conditions are necessary but not sufficient on their own. First, consider continuity. A function is continuous at a point if the limit as the input approaches the point from both sides exists and is equal to the value of the function at that point. If a function is not continuous, it cannot be differentiable at that point. For example, the piecewise function \\[ f(x) = \\begin{cases} 1 & \\text{if } x < 0 \\\\ 2 & \\text{if } x \\geq 0 \\end{cases} \\] has a jump discontinuity at \\( x = 0 \\). As the function approaches zero from the left, the value is 1; from the right, the value is 2. The limit does not exist at that point, so the function is not continuous and thus not differentiable at \\( x = 0 \\). Next, consider smoothness. A function may be continuous at a point but still fail to be differentiable if there is a sharp corner or cusp. The function \\( f(x) = |x| \\) is continuous everywhere, including at \\( x = 0 \\), but it is not differentiable at that point. This is because the slope of the graph changes abruptly from negative on the left to positive on the right. The left-hand derivative at \\( x = 0 \\) is -1, and the right-hand derivative is 1. Since the left-hand and right-hand derivatives are not equal, the derivative does not exist at that point. Thus, both continuity and smoothness are required for differentiability. A failure in either condition prevents a function from being differentiable.<\/p>\n\n\n\n In order to be differentiable, we must satisfy two conditions. A function must be continuous and smooth. Please give an example of a function that does not satisfy continuity and a function that does not satisfy smoothness. The Correct Answer and Explanation is: Answer: A function that does not satisfy continuity is:1 & \\text{if } […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-232629","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232629","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=232629"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232629\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=232629"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=232629"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=232629"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner9-333.jpeg\")
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