Suppose $f:[a,b]\to\mathbb R$ is a differentiable function with $f'\in L^1 [a,b]$, that is, $f$ has a derivative that is integrable on $[a,b]$. What are some properties $f$ must have?
If any are known, some defining properties would be nice, that is, properties $P$ such that $f$ has $P$ if and only if $f'$ is integrable on $[a,b]$.
One thing we can say is that $f$ must be Lipschitz-continuous. This shouldn't be the best we can do: Lipschitz-continuity follows from the boundedness of $f'$, but we know from Darboux's theorem that $f'$ also has the intermediate value property, a very restrictive property. I'm not sure how we could leverage this to get some insight into the nature of $f$. Any ideas?
Best Answer
If $f:[a,b]\to\mathbb R$ is differentiable everywhere then $f'\in L^1 [a,b]$ if and only if $f$ is absolutely continuous.
This follows from the equivalent definitions of absolute continuity and the following
which is Theorem 7.21 in Rudin's Real and Complex Analysis, see also Proofs of the second fundamental theorem of calculus on Math Overflow.
If $f$ is absolutely continuous then $f'$ is integrable on $[a, b]$. Conversely, if $f'$ is integrable, then $$ f(x) = f(a) + \int_a^x f'(t) \, dt $$ according to Rudin's theorem, and that implies that $f$ is absolutely continuous.