{"id":1258,"date":"2017-10-24T22:58:24","date_gmt":"2017-10-25T02:58:24","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1258"},"modified":"2017-10-24T22:58:24","modified_gmt":"2017-10-25T02:58:24","slug":"definitions-102417","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/24\/definitions-102417\/","title":{"rendered":"Definitions 10\/24\/17"},"content":{"rendered":"

During the lecture on Tuesday we continued our discussion of uniform continuity by discussing Theorem 4.4.7. This theorem seems as if it will be very useful in the future and thus I would like to dig deeper into it. <\/span><\/p>\n

1. Preliminaries<\/strong><\/p>\n

First let us recall definitions and theorems that will be mentioned.<\/span><\/p>\n

Compact Set<\/strong>: a set \"K is compact if every sequence \"x_n has a subsequence in \"K\" that converges to a limit \"l.<\/span><\/p>\n

\"\"<\/a>
\nUniform Continuity<\/strong>: A function \"f:A is uniformly continuous on \"A\" if for every \"\epsilon there exists a \"\delta such that for all \"x,y, \"\mid implies \"\mid.<\/p>\n

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

Theorem 4.4.5<\/strong> : A function\u00a0\"f:A is NOT uniformly continuous on \"A\" if there exists a \"\epsilon_0 and two sequences \"x_n\" and\u00a0\"y_n\" in\u00a0\"A\" satisfying\u00a0\"\mid but \"\mid.<\/p>\n

2. Uniform Continuity on Compact Sets<\/strong><\/p>\n

Theorem 4.4.7 builds on the topic of compact sets by stating an interesting observation about continuous functions on compact sets.<\/p>\n

Theorem 4.4.7:\u00a0<\/strong>A function that is continuous on a compact set \"K\" is uniformly continuous on \"K\".<\/p>\n

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

3. Proof in Steps<\/strong><\/p>\n

While the specifics of this proof were discussed in lecture, it would be valuable to recall the basic steps of the proof.<\/p>\n

Step 1:<\/strong> Negation of the Theorem. By negating the entire theorem, we are able to set up a proof by contradiction. The negation is as follows<\/p>\n

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

By assuming these three statement, one can begin a proof BWOC.<\/p>\n

Step 2:<\/strong>\u00a0Implement Theorem 4.4.5. Using this theorem, one can say that for some \"\epsilon_0, there exists two sequences \"x_n\" and\u00a0\"y_n\" in\u00a0\"K\" satisfying\u00a0\"\mid but \"\mid.<\/p>\n

Step 3:<\/strong>\u00a0Implement the definition of Compact. Since\u00a0\"x_n\" is in set \"K\", then there exists a convergent subsequence\u00a0\"x_nk\" such that\u00a0\"x_nk and \"x.<\/p>\n

Step 4<\/strong>: Implement Algebraic Limit Theorem. By once again using the definition of Compact, we can see that the subsequence\u00a0\"y_nk\" is a\u00a0convergent subsequence and its limit is an element of set\u00a0\"K\". However, by implementing the Algebraic Limit Theorem along with the assumption that\"\mid, one can see that \"y_nk.
\n\"\"<\/a><\/p>\n

Step 5: <\/strong>Recall the continuity of f. Since \"x_nk\" and \"y_nk\" both converge to x and we assumed that f is continuous at x, it must also be the case that \"lim(f(x_nk)-f(y_nk)). However, this presents a problem as we assumed that\u00a0\"\mid\u00a0for some \"\epsilon_0. With that, there exists a contradiction. Therefore, function that is continuous on a compact set must be uniformly continuous on that set.<\/p>\n

4. Conclusion<\/strong><\/p>\n

With Theorem 4.4.7 and its proof presented, one could ask about the importance of this theorem. I would argue that one good example of this was presented in the case of \"h(x) being uniformly continuous on the set \"[0,. By splitting the domain into sets \"[0, and \"[1,, we were able to use\u00a0Theorem 4.4.7 to quickly show that since \"h(x)\" is continuous on the compact set \"[0,,\u00a0\"h(x)\" must also be\u00a0uniformly continuous on the set. With this proven, we were then left with proving that \"h(x)\" is uniformly continuous\u00a0\"[1, and then going through the different cases regarding what set x and y are in. Remembering this example may be a good idea as implementing this technique of splitting the domain may prove to be useful in future proofs.<\/p>\n","protected":false},"excerpt":{"rendered":"

During the lecture on Tuesday we continued our discussion of uniform continuity by discussing Theorem 4.4.7. This theorem seems as if it will be very useful in the future and thus I would like to dig deeper into it. 1. Preliminaries First let us recall definitions and theorems that will be mentioned. Compact Set: a […]<\/p>\n","protected":false},"author":2684,"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":[58820,58823],"tags":[],"class_list":["post-1258","post","type-post","status-publish","format-standard","hentry","category-daily-blogs","category-definitions"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-ki","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1258","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\/2684"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1258"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1258\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1258"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1258"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1258"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}