How can I prove that no continuous function exists that satisfies the property of the dirac delta function? I thought it should be pretty easy, but it's actually giving me quite a hard time! I know that the integral of such a function must be 1, and that it must also be even (symmetric about the y-axis). It's also easy to see that such a delta function exists for any given function, but no such delta function exists for all functions. How do I go from here?