It is, in a way. The whole point of distributions is to extend the space of functions to one where more operations are permitted.

The limit of the Gaussian function as variance goes to 0 is not a function, but it is a distribution, the Dirac distribution.

Some distributions appear in intermediate steps while solving differential equations, and then disappear in the final solution. This is analogous to complex numbers sometimes appearing while computing the roots of a cubic function, but not being present in the roots themselves.