I am interested to what extent the famous identity

$\int_a^b f'(x) \ dx=f(b)-f(a)$

is true for a function $f:[a,b]\to \mathbb C$ continuous on $[a,b]$ and differentiable on $(a,b)$. One famous easy case of this problem is where $f'$ is continuous. In the above identity, the integral is with respect to Lebesgue measure on $\mathbb R$.

I have proven so far that $f'$ is always measurable on $(a,b)$ and that if $f'$ is bounded on $(a,b)$ then the result holds. The proof was reasonably elementary, making heavy use of the mean value theorem and the so-called bounded convergence theorem.

I felt that my condition was an artifact of the proof, as the bounded convergence theorem is considerably weaker than the dominated convergence theorem and its strengthened forms.

So does anyone know of a strengthened version of this result, or perhaps even a full description of all differentiable functions such that the above identity holds?

Thank you for your time and effort.