= Dirichlet-Jordan convergence theorem
{c}
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.
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