# Discontinuous Fourier transforms?

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

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Try $x \mapsto \mathbb{sinc}(x)$. – copper.hat May 11 '12 at 6:25
The answers and comments are sinc-ronizing. – copper.hat May 11 '12 at 6:57

Just take the inverse Fourier transform of your favourite discontinuous $L^2$ function.
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 $$\frac{\mathrm{sgn}(1+2x)+\mathrm{sgn}(1-2x)}{2}$$