{"id":1215,"date":"2017-10-22T19:17:52","date_gmt":"2017-10-22T23:17:52","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1215"},"modified":"2017-10-22T22:13:07","modified_gmt":"2017-10-23T02:13:07","slug":"hw6-challenge-1","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/22\/hw6-challenge-1\/","title":{"rendered":"HW6: Challenge 1"},"content":{"rendered":"

Assume that \"g\" is defined on an open interval \"(a,c)\" and it is known to be uniformly continuous on \"(a,b]\" and \"[b,c)\" where \"a<b<c\". Prove that \"g\" is uniformly continuous on \"(a,c)\".<\/p>\n

 <\/p>\n

 <\/p>\n

Assume \"g\" is defined on an open interval \"(a,c)\". Suppose \"g\" is uniformly continuous on \"(a,b]\" and \"[b,c)\" with \"a<b<c\".<\/p>\n

Suppose \"x\in. Since g is uniformly continuous on \"(a,b]\", we know that for any \"\epsilon>0\", \"\exists\delta_1>0\" s.t for \"x,b\in(a,b]\", \"\mid implies \"\mid.<\/p>\n

Similarly suppose \"y\in. Since \"g\" is uniformly continuous on \"[b,c)\", we know that for any \"\epsilon>0\", \"\exists\delta_2>0\" s.t. for \"y,b\in, \"\mid implies \"\mid.<\/p>\n

Choose \"\delta=\min\{\delta_1,\delta_2\}\" so that \"\mid implies \"\mid and \"\mid. Thus, applying the triangle inequality, for all \"x,y\in, if \"\mid, it follows that<\/p>\n

$\"\mid$<\/p>\n

Thus, \"g\" is uniformly continuous on \"(a,c)\" by definition.<\/p>\n","protected":false},"excerpt":{"rendered":"

Assume that is defined on an open interval and it is known to be uniformly continuous on and where . Prove that is uniformly continuous on .     Assume is defined on an open interval . Suppose is uniformly continuous on and with . Suppose . Since g is uniformly continuous on , we […]<\/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":[58827],"tags":[],"class_list":["post-1215","post","type-post","status-publish","format-standard","hentry","category-challenge-solutions"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-jB","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1215","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=1215"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1215\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1215"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1215"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}