{"id":1393,"date":"2017-11-02T13:15:44","date_gmt":"2017-11-02T17:15:44","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1393"},"modified":"2017-11-02T13:15:44","modified_gmt":"2017-11-02T17:15:44","slug":"definitions-103117","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/11\/02\/definitions-103117\/","title":{"rendered":"Definitions 10\/31\/17"},"content":{"rendered":"In class we discussed Darboux’s Theorem and the oddity that it implies. Differentiability of a function does not imply the continuity of its derivative, but it does imply the Intermediate Value Theorem holds for its derivative. For this blog, I would like to explore a function for which this oddity occurs.<\/p>\n
Consider the function ![\"g(x)=\begin{cases}](\"https:\/\/s0.wp.com\/latex.php?latex=g%28x%29%3D%5Cbegin%7Bcases%7D+x%5E2%5Csin%28%5Cfrac%7B1%7D%7Bx%7D%29%2C%26x%5Cneq+0%5C%5C0%2C%26+x%3D0%5Cend%7Bcases%7D&bg=ffffff&fg=000&s=0&c=20201002\")
discussed in the introduction to this chapter. As we have read, this function is differentiable in all of ![\"\mathbb{R}\"](\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BR%7D&bg=ffffff&fg=000&s=0&c=20201002\")
, but is not continuous at 0. What we have learned from Darboux’s Theorem, is that the Intermediate Value Theorem does apply to this function. This means that if you isolate an interval ![\"[a,b]\"](\"https:\/\/s0.wp.com\/latex.php?latex=%5Ba%2Cb%5D&bg=ffffff&fg=000&s=0&c=20201002\")
, then for any ![\"L\"](\"https:\/\/s0.wp.com\/latex.php?latex=L&bg=ffffff&fg=000&s=0&c=20201002\")
such that ![\"g'(a)<L<g'(b)\"](\"https:\/\/s0.wp.com\/latex.php?latex=g%27%28a%29%3CL%3Cg%27%28b%29&bg=ffffff&fg=000&s=0&c=20201002\")
or ![\"g'(b)<L<g'(a)\"](\"https:\/\/s0.wp.com\/latex.php?latex=g%27%28b%29%3CL%3Cg%27%28a%29&bg=ffffff&fg=000&s=0&c=20201002\")
, there exists ![\"c\"](\"https:\/\/s0.wp.com\/latex.php?latex=c&bg=ffffff&fg=000&s=0&c=20201002\")
such that ![\"g'(c)=L\"](\"https:\/\/s0.wp.com\/latex.php?latex=g%27%28c%29%3DL&bg=ffffff&fg=000&s=0&c=20201002\")
. With this function, this does make sense. Without including 0 in the interval, you just have a continuous function and the IVT holds like normal. If you do include 0 in the interval, that is where some ambiguity might arise. The problem is the derivative at 0. For this function ![\"g'(0)=0\"](\"https:\/\/s0.wp.com\/latex.php?latex=g%27%280%29%3D0&bg=ffffff&fg=000&s=0&c=20201002\")
. This is the one that we think the IVT might not hold for. But this function simply oscillates faster and faster as it approaches 0, so any interval you pick that contains 0 will still have an oscillation of the function such that we have a point so ![\"g'(c)=0\"](\"https:\/\/s0.wp.com\/latex.php?latex=g%27%28c%29%3D0&bg=ffffff&fg=000&s=0&c=20201002\")
.<\/p>\n","protected":false},"excerpt":{"rendered":"
In class we discussed Darboux’s Theorem and the oddity that it implies. Differentiability of a function does not imply the continuity of its derivative, but it does imply the Intermediate Value Theorem holds for its derivative. For this blog, I would like to explore a function for which this oddity occurs. Consider the function discussed […]<\/p>\n","protected":false},"author":3532,"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":[58823],"tags":[],"class_list":["post-1393","post","type-post","status-publish","format-standard","hentry","category-definitions"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-mt","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1393","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\/3532"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1393"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1393\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1393"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1393"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}