After all, it's only non-zero at one point.
closed as unclear what you're asking by user223391, user147263, Empty, Gyu Eun Lee, Najib Idrissi Sep 2 '15 at 8:37
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I think it has to do with the fact that continuity is implied by differentiability and integrability, and since the Dirac-Delta function is differentiable and integrable, it is continuous.