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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.
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June 18th, 2015, 08:34 AM   #2
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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.
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June 18th, 2015, 09:45 AM   #3
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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.
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June 19th, 2015, 08:43 AM   #4
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I don't understand what "g is continuous" has to do in this step. I mean if g is continuous, so?
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June 19th, 2015, 09:36 AM   #5
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If $g$ is not continuous then $\Delta u$ does not necessarily go to zero and the first term is no longer a difference quotient.
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June 19th, 2015, 09:40 AM   #6
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I don't understand the concept of continuous, can you explain more?
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June 19th, 2015, 10:09 AM   #7
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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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