Suppose $f: [0, 1] \rightarrow \mathbb{R}$ is continuous, and is continuously differentiable on $(0, 1)$. Can we conclude that $f$ is absolutely continuous over $[0, 1]$? On the one hand, we clearly know that $f$ is absolutely continuous on any closed interval in $(0, 1)$. But this seems not enough to guarantee that $f$ is absolutely continuous on the entire interval. On the other hand, let $\int_0^x f'(t) \mathrm{d}t$ be the improper intergal $\lim_{y \downarrow 0} \int_y^x f'(t) \mathrm{d}t$, which exists since $f$ is continuous at $0$. Then, the formula $f(x) = f(0) + \int_0^x f'(t) \mathrm{d}t$ holds for all $x$. I don't know how to reconcile these two observations.
Thank you!
The function $f(x)=x\sin(1/x)$ (with $f(0)=0)$ is continuously differentiable on $(0,1]$, but not absolutely continuous on $[0,1]$.