Differentiable almost everywhere of antiderivative function

Question

Give an integrable function $h$ on $[a,\,b]$. Let $f(x) = \int_{a}^{x} h$ for all $x\in [a,\,b]$. Prove that $f$ is differentiable almost everywhere on $(a,\,b)$\

My attempt: I tried to show that $f$ is differentiable on any subinterval $[c,d]\subset (a,b)$, but I could not see how to use the assumption $h$ is integrable despite spending several hours. Can anyone please help me with this problem?

Answer 1

Note that

$$h = h^+ - h^-,$$

where

$$h^+ = \max(h,0) \geqslant 0, \\ h^- = \max(-h,0) \geqslant 0. $$

Then

$$f(x) = \int_a^x h^+ - \int_a^xh^-.$$

Since $f$ is the difference of two monotone increasing functions, it is of bounded variation and is differentiable almost everywhere.

Answer 2

Another way to prove it is to note that:

$$m(b-a)\le \int_a^b f(x)\ dx \le M (b-a)$$

Where $m,M$ are inf/sup of $f(x)$. Thus:

$$|f(b)-f(a)|\le C |b-a|$$

So $f$ is Lipschitz and thus differentiable a.e.