Fourier series of a periodically extended quadratic arch ($f_{\mathrm{per}}(x)=\frac{1}{6}-\frac{1}{{\pi }^2}{\sum }_{n\ge 1}\frac{\cos (2\pi nx)}{n^2}$)

The period-one extension of $x(1-x)$ on $[0,1]$ has Fourier series $1/6-{\pi }^{-2}{\sum }_{n\ge 1}{n}^{-2}\cos (2\pi nx)$. Endpoint agreement makes the periodic function continuous. Its derivative series is $(2/\pi ){\sum }_{n\ge 1}{n}^{-1}\sin (2\pi nx)$, which equals $1-2x$ in the open unit interval and converges to zero at integers, the midpoint of the derivative's jump. Termwise differentiation of a Fourier series applies locally away from the corners.