Generalize function to allow adding some useful things which people wanted to be classical functions but which are not,

It therefore requires you to redefine and reprove all of calculus.

For this reason, most people are tempted to assume that all the hand wavy intuitive arguments undergrad teachers give are true and just move on with life. And they generally are.

One notable example where distributions pop up are the eigenvectors of the position operator in quantum mechanics, which are given by Dirac delta functions, which is most commonly rigorously defined in terms of distribution.

Distributions are also defined in a way that allows you to do calculus on them. Notably, you can define a derivative, and the derivative of the Heaviside step function is the Dirac delta function.