While solving the Integral $$ \int f(g(x))g'(x)dx$$ we substitute $g(x)=u$ and the integral becomes $$ \int f(u)du$$ but the condition is that $g(x)$ must be a differentiable function and $f(x)$ must be continuous on the range of $g(x)$ but in some books it is written that $g(x)$ must be a continuously differentiable function. So I want to know that which condition is sufficient and why?
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See https://math.stackexchange.com/questions/930780/is-a-differentiable-function-always-continuous – jl00 Sep 21 '19 at 15:19
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My question is not about are differentiable functions continuous – user679770 Sep 21 '19 at 15:23
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1If $F$ is an antiderivative of $f$, i.e. $F'=f$, then by chain rule, $F(g(x))$ is going to be an antiderivative of $f(g(x))g'(x)$ as long as $F,g$ are differentiable. This happens when $f$ is continuous and $g$ is differentiable. So, I really don't see why you need "continuous differentiability". – Julian Mejia Sep 21 '19 at 15:25
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You can weaken the requirements on $f$ by strengthening the requirements on $g$, and vice versa. By assuming that $g'$ is continuous, you don't need to assume that $f$ is continuous. Since practically every differentiable function of note has a continuous derivative (it is actually a bit hard for a derivative to not be continuous), pushing the continuity onto $g'$ might be considered the more useful version. – Paul Sinclair Sep 21 '19 at 23:52