Discontinuous Fourier transforms?

Question

What's an example (or even better a large class of examples) of an $L^2$ function whose Fourier transform is discontinuous?

Answer 1

Just take the inverse Fourier transform of your favourite discontinuous $L^2$ function.

Answer 2

Robert Israel gives the most general answer, but here is an explicit example.

By scaling this answer, it is shown that the Fourier Transform of the $\mathrm{sinc}$ function $$ \mathrm{sinc}(x)=\frac{\sin(\pi x)}{\pi x} $$ is the square bump function $$ \frac{\mathrm{sgn}(1+2x)+\mathrm{sgn}(1-2x)}{2} $$