I want help in showing that $f$ is Lipschitz on $[0,1]$ $\implies$ that $f$ can written in the form $$f(x) = f(0) + \int_0^x h(x)~dt$$ for some bounded Lebesgue measurable function $h$ on $[0,1]$.
$f$ being Lipschitz on $[0,1]$ implies there is some constant $K$ such that $|f(x)-f(y)|\leq K |x-y|$ for every $x,y \in [0,1]$.
Can I argue as follows:
I know that if $f$ is Lipschitz on $[0,1]$, then $f$ is abosolutely continuous on $[0,1]$ and so $f$ is a definite integral. i.e. $$f(x) = f(0) +\int_0^x f'(t)~dt,$$ So I can take $f' = h$. Can I do this or there is a better way of approaching it.