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Let's say that I'm integrating a composite function, say $f(g(x))$, that is in a form to which I can apply the substitution rule. Is it true to say that both $f$ and $g$ must be differentiable?

I understand that the substitution rule requires $g$ to be differentiable and that the substitution rule relies on the chain rule, and the chain rule requires both f and g to be differentiable.

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    Suppose we want to do $\int_0^1 e^{-1/x}dx$. With $u=1/x$, this becomes $\int_1^\infty e^{-u}u^{-2}du$. The function $g(x)=1/x$ is nondifferentiable at $x=0$, but this isn't a problem. The problem here may be that you probably have a certain theorem in mind, and you want to know whether a certain assumption in the theorem is really necessary -- but you haven't stated the theorem.2012-02-07

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