Dirichlet's theorem on Fourier series ($S_{N}f(x)\to [f(x-)+f(x+)]/2$)

One standard sufficient form states that the Fourier series of a piecewise continuously differentiable periodic function converges at $x$ to $[f(x-)+f(x+)]/2$. At a continuity point it equals $f(x)$. This modern sufficient form should be distinguished from the exact hypotheses of the 1829 original.^[ref]