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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
up vote 3 down vote accepted

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

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for example, this is not hard to compute for a characteristic function – user12014 May 11 '12 at 6:42

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 $$ \frac{\mathrm{sgn}(1+2x)+\mathrm{sgn}(1-2x)}{2} $$

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