Does N-L formula holds for everywhere differentiable function?

Question

Suppose that $f$ is differentiable on $[a,b]$ everywhere, if $f'(x)$ is Lebesgue integrable on $[a,b]$, can we say that the Newton-Leibniz formula holds for $f$?, More precisely, dose the following holds? $$f(b)-f(a)=\int_a^bf'(x)d x.$$ Note that if everywhere has been replaced by a.e., then then Cantor-Lebesgue function is a counterexample.

Answer 1

Yes, it is well-known theorem, and you can find it in I.P.Natanson, Theory of functions of a real variable, Chapter IX, the last paragraph, theorem 1. It states as follow.

If the derivetive $f'(x)$ exists everywhere, $x\in[a,b]$ , is finite, and is summable, then for any $x\in[a,b]$ holds $$f(x)=f(a)+\int_a^x f'(t) \, dt.$$