June 18th, 2015, 08:02 AM  #1 
Newbie Joined: Jun 2015 From: Hong Kong Posts: 4 Thanks: 0  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.

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: 7,659 Thanks: 2635 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: 7,659 Thanks: 2635 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: 7,659 Thanks: 2635 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. 

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