I want to prove that: If $f(x)$ is continuous with a period of $2\pi$ and its derivative $f^\prime(x)$ is piecewise continuous, then the Fourier series of $f(x)$ converges uniformly to $f(x)$.
I'm familiar with the exact same proof (presented below) except for the fact that $f^\prime(x)$ has jump discontinuities.
Don't these discontinuities affect the uniform convergence of $f$ ? What changes in the proof if $f^\prime$ is piecewise continuous?
The proof works all the same, with the caveat the definition of "piecewise smooth" only allows discontinuity jumps, and thus $f'\in L^2$.
At the very end of the proof, it is not obvious to me that $f$ is of bounded variation, but one can use the result that if $f$ is continuous and its Fourier series converges uniformly, then it equals $f$.