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June 18th, 2015, 08:02 AM   #1
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Proof of chain rule questions

Dear all, I read a proof of chain rule from a book but I don't understand why the statement "delta u -> 0 as delta x -> 0 since g is continuous" makes the proof as attached below. Could anyone answer me for this? Thanks a lot.
Attached Images 20150618_220036.jpg (88.3 KB, 8 views) June 18th, 2015, 08:34 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 I presume that in the part of the page you did not copy, u is define by u= g(x). Further $\displaystyle \Delta u= u(x+h)- u(x)= g(x+h)- g(x)$. The definition of "g is continuous" is that so the limit of g(x+h)- g(x), as h goes to 0, is g(x)- g(x)= 0. June 18th, 2015, 09:45 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra The reason for stating this is that the limit variable has been changed from $\Delta x$ to $\Delta u$. We need to know that the doesn't change the value of the limit. Note at the bottom of the page is extremely important. June 19th, 2015, 08:43 AM #4 Newbie   Joined: Jun 2015 From: Hong Kong Posts: 4 Thanks: 0 I don't understand what "g is continuous" has to do in this step. I mean if g is continuous, so? June 19th, 2015, 09:36 AM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra If $g$ is not continuous then $\Delta u$ does not necessarily go to zero and the first term is no longer a difference quotient. June 19th, 2015, 09:40 AM #6 Newbie   Joined: Jun 2015 From: Hong Kong Posts: 4 Thanks: 0 I don't understand the concept of continuous, can you explain more? June 19th, 2015, 10:09 AM #7 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra A function $f(x)$ is continuous at the point $x=a$ if the following (two sided) limit relation holds: $$\lim_{x \to a} f(x) = f(a)$$ More informally, a function is continuous if you can draw it without taking your pen off the paper. Or, equivalently, if it doesn't "jump" from one value to another. Tags chain, proof, questions, rule prove chain rule mathsforum

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