Let $f:[a,b]\to \mathbb{R}$ be differentiable everywhere on $(a,b)$ and has an integrable derivative, is f absolutely continuous?

Question

Let $f:[a,b]\to \mathbb{R}$ be differentiable everywhere on $(a,b)$ and have integrable derivative, is f absolutely continuous?

There is a theorem in Rudin that says this is true if f is differentiable on $[a,b]$, can we loosen this to differentiable on $(a,b)$?