Dirac Delta special case

Question

How can we define a generalized function, very similar with Dirac Delta, that is zero on the real line but 1 at 0 (origin), instead of "infinity"?

Something without using piecewise notation?

I'm looking for something that I can use with Mathematica to inspect some properties on the real numbers.

If possible I would some help to define this function on Mathematica too.

Answer 1

Mathematica has a function called DiscreteDelta for such purposes:

DiscreteDelta[0] gives $1$; DiscreteDelta[n] gives $0$ for other numeric $n$.