{"id":1218,"date":"2017-10-22T17:54:53","date_gmt":"2017-10-22T21:54:53","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1218"},"modified":"2017-10-22T17:58:52","modified_gmt":"2017-10-22T21:58:52","slug":"what-happened-1019","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/22\/what-happened-1019\/","title":{"rendered":"What Happened 10\/19"},"content":{"rendered":"

On Thursday 10\/19, we began class with a weekly situation, “Are we doing mathematics or philosophy?”. Dr. LeCrone then read a thorough and quite verbose definition of a function from another analysis book. We then switched to the topic of the day: continuity, part of Ch. 4 (Continuity on Compact Sets and Uniform Continuity).<\/p>\n

We learned the definition of uniform continuity and point-wise continuity. We then looked at three example functions, \"f(x)= all with \"A=[0,\infty)\", and examined their graphs, specifically examining \"V_{\delta_1} and \"V_{\epsilon_1}. Through this, we were able to visually apply our definition of uniform continuity and check whether each met the definition.<\/p>\n

Dr. LeCrone then gave a proof template, and we completed the proof for proving \"f(x)\" was continuous on the domain.<\/p>\n

Proof template: let \"\epsilon>0\" … (details regarding \"\delta\") … choose \"\delta=\" ____ , depending only on \"\epsilon\". Let \"c be arbitrary and consider \"x with \"|x-c| … (details).. Thus, \"|f(x)-f(c)|<. Since \"c\" was arbitrary, we have unif. continuity of \"f\" on domain \"A\".<\/p>\n

We then completed this proof, using \"\delta=<\/p>\n

We then constructed a Sequential Criteria for Uniform Continuity and and a Sequential Characterization.<\/p>\n

We used this sequential characterization, choosing \"(x_n)=(n),, and \"\epsilon_0=1\" to show that \"g(x)\" is not uniformly continuous.<\/p>\n

We observed that \"h(x)\" requires us to split the domain in order to characterize, which we will cover on Tuesday.<\/p>\n

Finally we covered two theorems. Theorem 4.4.1 states that If $latex\u00a0f: A \\rightarrow R$ continuous and \"K is compact, then \"f(k)\" is compact. We went over a pf of this. Theorem 4.4.2 states that if $latex\u00a0f: K \\rightarrow R$ is continuous on \"K compact, then \"f\" attains a maximum and minimum on \"K\". We then converted this to two equivalent statements, one relating values of \"f(x)\" directly and another using supremum and infimum.<\/p>\n","protected":false},"excerpt":{"rendered":"

On Thursday 10\/19, we began class with a weekly situation, “Are we doing mathematics or philosophy?”. Dr. LeCrone then read a thorough and quite verbose definition of a function from another analysis book. We then switched to the topic of the day: continuity, part of Ch. 4 (Continuity on Compact Sets and Uniform Continuity). We […]<\/p>\n","protected":false},"author":3527,"featured_media":0,"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":[58820,58821],"tags":[],"class_list":["post-1218","post","type-post","status-publish","format-standard","hentry","category-daily-blogs","category-what-happened-today"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","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\/1218","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\/3527"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1218"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1218\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1218"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1218"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1218"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}