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It seems like the Dirac Delta function is discontinuous as it has a value of $\infty$ at $x=0$ and $0$ everywhere else. It looks to be same as the Kronecker Delta Function, which we know to be discrete.

Anirudh Kalla
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Jeff Strom
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    Continuous is relative to a pair of topologies. It is continuous as a function from $C^\infty_c$ to $\mathbb{R}$. – Ian Sep 02 '15 at 00:04
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    In what context do people call it continuous? – littleO Sep 02 '15 at 00:11
  • Wikipedia does at the end of this section. I can see why it might be considered to be continuous, but it doesn't quite make sense when there's only one possible value in the range – Jeff Strom Sep 02 '15 at 00:14
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    You should not think of the Dirac delta as a function of a real variable at all; the oft-quoted formula "$\delta(0)=+\infty,\delta(x)=0$ otherwise" has no real mathematical content. The valid arguments of the Dirac delta are test functions, not points. – Ian Sep 02 '15 at 00:21
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    Wikipedia is using the term "continuous" very loosely in that example. They are using it just to mean, like, "defined for all real numbers, as opposed to just defined on the integers". (But even that statement is not really correct, because technically the delta function is a distribution.) – littleO Sep 02 '15 at 02:23
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    Isn't the constant zero function continuous? – Carl Mummert Sep 02 '15 at 02:24

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