{"id":1192,"date":"2017-10-23T13:45:23","date_gmt":"2017-10-23T17:45:23","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1192"},"modified":"2017-10-23T13:45:23","modified_gmt":"2017-10-23T17:45:23","slug":"hw-6-challenge-2-ab","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/23\/hw-6-challenge-2-ab\/","title":{"rendered":"HW 6 Challenge 2 a,b"},"content":{"rendered":"

Consider the function \"f(x)=\frac{1}{x}\".<\/p>\n

\"\"<\/a><\/p>\n

We will consider the domains \"(0,1)\" and \"[1,\infty)\".<\/p>\n

A)\u00a0<\/strong>To prove that\u00a0\"f(x)=\frac{1}{x}\" is not uniformly continuous on\u00a0\"(0,1)\", we need to show that for some \"\epsilon_\circ>0\", there exist sequences \"(x_n), such that \"|x_n-y_n|\rightarrow, but\u00a0\"|f(x_n)-f(y_n)|\geq.<\/p>\n

Let \"(x_n):=\{x_n\in and\u00a0\"(y_n):=\{y_n\in for \"n\in. Notice that \"|x_n-y_n|=|\frac{2n+1}{2^n}-\frac{2n}{2^n}|=|\frac{1}{2^n}|\rightarrow. Yet \"|f(x_n)-f(y_n)|=|\frac{2^n}{2n+1}-\frac{2^n}{2n}|=|\frac{2^n}{2n(2n+1)}|>0\" for all\u00a0\"n\in. Hence by the archimedean property we can choose \"0<\epsilon_\circ<|\frac{2^n}{2n(2n+1)}|\". Thus\u00a0\"f(x)=\frac{1}{x}\" is not uniformly continuous on\u00a0\"(0,1)\".<\/p>\n

B)<\/strong> To prove that\u00a0\"f(x)=\frac{1}{x}\" is uniformly continuous on\u00a0\"[1,\infty)\", we need to show that for every \"\epsilon>0\", there exists a \"\delta>0\" such that for all \"x,c\in\u00a0[1,\infty)\", \"|x-c|<\delta\" implies \"|f(x)-f(c)|<\epsilon\".<\/p>\n

Let\u00a0\"\epsilon>0\" be arbitrary. Choose a \"\delta<\epsilon\". Let \"c\in be arbitrary and consider \"x\in with \"|x-c|<\delta\". Thus \"|f(x)-f(c)|=|\frac{1}{x}-\frac{1}{c}|=|\frac{x-c}{xc}|\leq since we have\u00a0\"xc\geq.<\/p>\n

Hence \"f(x)=\frac{1}{x}\" has uniform continuity on\u00a0\"[1,\infty)\" because \"c\" was arbitrary.<\/p>\n","protected":false},"excerpt":{"rendered":"

Consider the function . We will consider the domains and . A)\u00a0To prove that\u00a0 is not uniformly continuous on\u00a0, we need to show that for some , there exist sequences such that , but\u00a0. Let and\u00a0 for . Notice that . Yet for all\u00a0. Hence by the archimedean property we can choose . Thus\u00a0 is […]<\/p>\n","protected":false},"author":3528,"featured_media":1193,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[58827],"tags":[],"class_list":["post-1192","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-challenge-solutions"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/blog.richmond.edu\/math320\/files\/2017\/10\/1x.jpg?fit=579%2C570&ssl=1","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-je","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1192","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/users\/3528"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1192"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1192\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media\/1193"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1192"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1192"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1192"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}