pointwise convergence (of a Fourier series)

What a Fourier series adds up to at one specific x -- the function's own value where f is continuous, and the average of the two one-sided limits at a jump.

At a point $x$ where $f$ is continuous (and piecewise smooth nearby), $f$ 's Fourier series converges to $f(x)$ itself; at a jump discontinuity, it converges instead to the average of the left- and right-hand limits, $\tfrac12\big[f(x^-)+f(x^+)\big]$ .