{"id":1542,"date":"2017-11-20T21:35:24","date_gmt":"2017-11-21T02:35:24","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1542"},"modified":"2017-11-26T12:34:07","modified_gmt":"2017-11-26T17:34:07","slug":"muddiest-point-1116","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/11\/20\/muddiest-point-1116\/","title":{"rendered":"Muddiest Point 11\/16"},"content":{"rendered":"

In this edition of the muddiest point I will go over non-Riemann integrable functions. Let’s review a classic example that we mentioned in class, the Dirichlet function. Define the Dirichlet Function as
\n$\"f(x)=\begin{cases}$
\nRecall that by definition, A bounded function \"f\" defined on the interval \"[a,b]\" is Riemann-integrable if \"U(f)=L(f)\". For \"\mathbb{P}\", the set of all possible partitions of \"[a,b]\", we call \"U(f)=\inf\{U(f,P):P\in\mathbb{P}\}\" the upper integral of \"f\". Similarly, \"L(f)=\sup\{L(f,P):P\in\mathbb{P}\}\" is the lower integral of \"f\". (Notice that these use the lower and upper sums that we discussed in class. We define the lower sum \"L(f,P)=\sum_{k=1}^n and the upper sum \"U(f,P)=\sum_{k=1}^n).<\/p>\n

Well, looking at the Dirichlet function, we can see that \"U(f)=1\", but \"L(f)=0\". It follows that \"1\neq, so the function is not Reimann integrable. This way of showing that the upper and lower integrals of a functions are not equal will most likely be one of our main tools to show that a function is not Riemann integrable.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this edition of the muddiest point I will go over non-Riemann integrable functions. Let’s review a classic example that we mentioned in class, the Dirichlet function. Define the Dirichlet Function as $$ Recall that by definition, A bounded function defined on the interval is Riemann-integrable if . For , the set of all possible […]<\/p>\n","protected":false},"author":3526,"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":[58822],"tags":[],"class_list":["post-1542","post","type-post","status-publish","format-standard","hentry","category-muddiest-point"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-oS","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1542","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\/3526"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1542"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1542\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1542"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1542"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1542"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}