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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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    The answers and comments are sinc-ronizing.2012-05-11

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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 function2012-05-11
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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} $