What conditions are necessary for Integration By Substitution?

Question

My book says following on integration by substitution:

Theorem: Let $f$ and $g$ be differentiable functions and $g'$ be continuous. Then $$\int f(g(x))\cdot g'(x) dx=F(g(x))+c,$$ where $F$ is antiderivitive of $f$. Proof: $$(F(g(x))+c)'= (F(g(x))'+(c)'=f(g(x))\cdot g'(x).$$

Where in the proof I need the assumption that $g'$ is continuous? Is it correct that chain rules cannot be used if my functions $f$ and $g$ are not differentiable? In some books they just assume that $f$ is continuous and $g$ is differentiable. What should the necessary assumptions in this theorem be?

Answer 1

The change-of-variable theorem, as written, refers to indefinite integration, so the requirements that $f$ be differentiable and $g'$ be continuous are indeed extraneous. (In fact, they can be loosened even in the definite-integration version of the theorem.) If being extremely rigorous, for the stated result, we do require that $g$ has an interval domain $D,$ and that $F$ is the antiderivative of $f$ on $g[D].$

Here is a better statement of the theorem, as well as the definite-integration version, where all the conditions are truly necessary.