What you're looking for is the Dirac delta "function"; specifically
$f(x) = \delta(x-c).$
I've put "function" in scare quotes above, since the Dirac delta is not actually a real function, although it can in many cases be treated as one.
Informally, you can visualize $\delta(x)$ as a function which has an infinitely tall peak at $x=0$ and is zero everywhere else; specifically, the "infinitely tall peak" needs to be just tall enough to have the area under it integrate to $1$. Of course, no actual real-valued function can have such a peak, but it turns out that, if you simply pretend that such a function exists and don't ask any hard questions about what its value at $x=0$ actually is, many calculations will just work as if nothing odd was going on.
Formally, the Dirac delta can be defined as a generalized function, specifically as a distribution. (And no, the similarity of the name with "probability distribution" is not a coincidence.)