 Dirichlet-Jordan convergence theorem

The Fourier series of a periodic function of bounded variation converges at each point to the mean of its two one-sided limits. In particular, it converges to the function at continuity points. Periodizing a compact interval introduces jumps at its integer endpoints; their half-values explain the half-weight convention in the Van der Corput sum-integral lemma. The conclusion is pointwise convergence, not a claim of absolute convergence of every such Fourier series.