Distributions were introduced to model certain concepts from Physics. Physics has the concept of a "mass of points in space". The Dirac delta function can model an electromagnetic charge of a point in space. The Dirac delta function is zero everywhere, except at one point, where it is infinitely large. This needs to be the case, because the density function needs to be 1. There is no function that can meet this criterion, except if integraton is taken as a "function" in the mathematical sense.