I know that if a function is everywhere differentiable on $[0,1]$ it need not be absolutely continuous. Also a function that is almost everywhere differentiable, continuous and non-decreasing may fail to be absolutely continuous. But what if $f$ is differentiability everywhere and non-decreasing? Are these enough to guarantee I can use the Fundamental Theorem of Calculus? I'm grateful for any references you might have.
If $f$ is nondecreasing on $[0,1]$ then $f$ has bounded variation and $f' \in L^1([0,1])$. If such a function is differentiable everywhere in $[0,1]$ then it satisfies $$f(x) = f(0) + \int_0^x f'(t) \, dt$$ for all $x \in [0,1]$ and is consequently absolutely continuous.
I don't have a short proof handy, but this is Theorem 7.21 in Real and Complex Analysis, 3rd ed. by Rudin.