{"id":1655,"date":"2017-12-06T14:34:21","date_gmt":"2017-12-06T19:34:21","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1655"},"modified":"2017-12-06T14:34:21","modified_gmt":"2017-12-06T19:34:21","slug":"muddiest-point-1205","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/12\/06\/muddiest-point-1205\/","title":{"rendered":"Muddiest Point 12\/05"},"content":{"rendered":"

I think the muddiest point in today’s lecture was the relation between continuity, differentiability. The reason why I pick this point was that it is very easy to use the false conclusion when we are working on problems. In this post, I would start with the theorem in the textbook and then use graphs\u00a0to provide some counterexamples intuitively.<\/p>\n

We know that differentiability implies contnuity, as stated in the textbook.<\/p>\n

Theorem 5.2.3. If \"g is differentiable at a point \"c\in, then g is continuous at c as well.<\/p>\n

But continuity does not imply differentiability:
\nThese are some simple examples of integrable functions that are not continuous<\/p>\n

Example 1:\u00a0\"f(x)=|x|\" is continuous at 0, but is not differentiable at 0.<\/p>\n

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

The absolute value function \"f(x)=|x|\" is continuous at 0, but is not differentiable at 0. Intuitively, the graph of the absolute value function has a “sharp point” at the origin. Thus, the absolute value function is continuous at 0 but is not differentiable at 0.<\/p>\n

 <\/p>\n

Example 2: \"f(x)=x^\frac{2}{3}\" \u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>is continuous at\u00a00<\/span><\/span><\/span><\/span><\/span><\/span><\/span>, but not differentiable at 0. Similar to Example 1, there is a “sharp point” at 0.<\/p>\n

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

Example 3: \"f(x)=\sqrt[3]{x}\"\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0is continuous at\u00a00<\/span><\/span><\/span><\/span><\/span><\/span><\/span>, but not differentiable at 0.The tangent line at\u00a00<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0is vertical. The derivative (slope) is undefined for a vertical line, so the derivative does not exist at 0.<\/p>\n

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

I think the muddiest point in today’s lecture was the relation between continuity, differentiability. The reason why I pick this point was that it is very easy to use the false conclusion when we are working on problems. In this post, I would start with the theorem in the textbook and then use graphs\u00a0to provide […]<\/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":[58822],"tags":[],"class_list":["post-1655","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-qH","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1655","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=1655"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1655\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1655"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1655"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1655"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}