{"id":1179,"date":"2017-10-22T00:35:12","date_gmt":"2017-10-22T04:35:12","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1179"},"modified":"2017-10-22T14:49:05","modified_gmt":"2017-10-22T18:49:05","slug":"muddiest-point-101917","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/22\/muddiest-point-101917\/","title":{"rendered":"Muddiest Point 10\/19\/17"},"content":{"rendered":"

I think the muddiest point from Thursday\u2019s class is the distinction between Uniform continuity and point-wise continuity.<\/p>\n

First, we recall the definitions of pointwise continuity and uniform continuity.<\/p>\n

Uniform continuity: A function f : A \u2192 R is uniformly continuous on A if for every \u03b5 > 0 there exists a \u03b4 > 0 such that for all x, y \u2208 A, |x \u2212 y| < \u03b4 implies |f(x) \u2212 f(y)| < \u03b5.<\/p>\n

Point-wise continuity: A function f : A \u2192 R is continuous at a point c\u2208A if,for all \u03b5>0,there exists a \u03b4>0 such that whenever |x\u2212c|<\u03b4 (and x \u2208 A) it follows that |f(x) \u2212 f(c)| < \u03b5.<\/p>\n

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With uniform continuity, the choice for\u00a0\u03b4 depends only on\u00a0\u03b5 whereas when showing a function is continuous at a point c. the choice for\u00a0\u03b4 is allowed to depend on both\u00a0\u03b5 and c.<\/p>\n

To better visualize the difference, I would like to demonstrate them in images.<\/p>\n<\/div>\n

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

Point-wise continuity: A function is continuous if, for a given point p0, for any\u00a0\u03b5, we can find some\u00a0\u03b4, such that the distance between the points (p0 and p in the picture)must be less than\u00a0\u03b4 and the distance of images is less than\u00a0\u03b5.<\/p>\n

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

Uniform continuity: We need to guarantee the two points in the\u00a0\u03b4 range, the images of the two points will be fall in the\u00a0\u03b5 range. For uniform continuity, we have to show this for every two points in the function.<\/p>\n

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In addition, uniform continuity is a strictly stronger property.\u00a0If f is \u201cuniformly continuous on A\u201d, \u03b5 and \u03b4 can be chosen that works simultaneously for all points c in A.<\/p>\n

If a function f is not uniformly continuous on a set A, it tells us that there is some \u03b5 > 0 for which no single \u03b4 > 0 satisfies\u00a0that the images of the two points fall into the\u00a0\u03b5 range for all c \u2208 A.<\/p>\n

 <\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

I think the muddiest point from Thursday\u2019s class is the distinction between Uniform continuity and point-wise continuity. First, we recall the definitions of pointwise continuity and uniform continuity. Uniform continuity: A function f : A \u2192 R is uniformly continuous on A if for every \u03b5 > 0 there exists a \u03b4 > 0 such […]<\/p>\n","protected":false},"author":2107,"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":[58820,58822],"tags":[],"class_list":["post-1179","post","type-post","status-publish","format-standard","hentry","category-daily-blogs","category-muddiest-point"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-j1","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1179","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\/2107"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1179"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1179\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1179"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1179"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}