Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 4 / a / Solution

The graph repeats every two units: it rises linearly from the right-hand limit $-1$ at an odd integer to value one at the next even integer, then falls linearly to value zero at the next odd integer. The defined value at each odd integer is zero, while its right-hand limit is minus one.

The Dirichlet convergence theorem for Fourier series gives the midpoint of the two one-sided limits at a jump. Hence, on the requested interval, the Fourier sum differs from $f$ precisely atAt every other point it equals $f(x)$; in particular it equals one at the continuous corners with even integer coordinates.