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September 18th, 2011, 07:16 PM   #1
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proof of the chain rule - one thing I don't understand

I have a question about the proof of the chain rule as it appears on wikipedia (http://en.wikipedia.org/wiki/Chain_rule#First_proof).

I understand that if g(x)-g(a)=0 then one step would be multiplying by 0/0, but I don't understand why - to remedy this - we can't just add another given that "as x-a tends to 0, if g(x)-g(a)=0, there exists a smaller x-a such that g(x)-g(a) is nonzero".

I can think of two reasons that this wouldn't be satisfactory in several cases. One is that if g is 0 on an interval, then our proof of the chain rule wouldn't hold, and of course the way I did it, we'd have to add another given (saying that there is always a smaller nonzero g(x)-g(a)). However, wouldn't what I said be valid for functions where g(x)=x*sin(1/x)?
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September 21st, 2011, 04:37 AM   #2
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See this discussion (post #6 in particular, but beware of typing errors in it).
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