All the theorems about substitution rule for definite integrals that I come across state, that if $f$ is a continuous function on $I$ and if $\phi$ transforms $[a,b]$ into $I$ and is continuously differentiable then $$\int_{\phi(a)}^{\phi(b)}f(x)dx=\int_{a}^{b}f(\phi(t))\phi'(t)dt$$ But only one source that I foundstates, that $f$ only has to be integrable in it's domain for the rule to apply. Is it true?
I'm asking because I want to use the subtitution rule in one of the theorems I'm trying to prove, but the only thing I know about my $f$ is that it's integrable, not that it's continuous and therefore the second version of subtitution rule would satisfy me, not the first one.
