Does piecewise continuity of $f'$ imply piecewise continuity of $f$?

Question

In stating Dirichlet's theorem on Fourier convergence, it seems standard to assume that both $f$ and $f'$ are piecewise continuous. Is it possible that piecewise continuity of $f'$ implies piecewise continuity of $f$ ?

Answer 1

Differentiability implies continuity (pointwise). So if we are saying that $f'$ is piecewise continuous, i.e. continuous on the open intervals $(a_i, a_{i+1})$ for some finite number of points $a_1, \dots, a_n$, then we are in particular asserting that $f'$ exists on those intervals. Thus $f$ is continuous on those same open intervals.

Moreover, if $f'$ is continuous on some $(a, a+\epsilon]$ with a finite right-side limit at $a$, then in particular it is bounded in absolute value on that interval by some number $M$. By the mean value theorem, $f$ is thus $M$-Lipschitz on $(a, a+\epsilon]$ and so it is uniformly continuous on that interval. This implies that $f|_{(a, a+\epsilon]}$ has a (uniformly) continuous extension to $[a, a+\epsilon]$, which is the same as saying that the one-sided limit at $a$ exists. (You can reduce that general statement to an elementary calculus argument if you wish.)