= Solution
{c}
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 at
$$
\boxed{x\in\{-3,-1,1,3\},\qquad\text{Fourier sum}=-\frac12\ne f(x)=0.}
$$
At every other point it equals $f(x)$; in particular it equals one at the continuous corners with <even integer> coordinates.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-v-periodic-fourier.png]
{title=The periodic piecewise linear function and its Fourier midpoint values at jumps}
{description=Filled black circles are the assigned odd-integer values, hollow circles show the right-hand limits, and blue diamonds give the Fourier sum at each jump.}
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