Distributions are sometimes called ‘generalized functions’, and that is essentially what they are. They correspond to situations presented to us by physical experience which are not adequately covered by the traditional y = f(x) notion of a function. An example is the well-known Dirac Delta Function, which is in fact not a function in the standard sense. The Dirac ‘function’ corresponds to a unit impulse imparted to a system over what we may idealize as an infinitely short interval of time. Think, for example, of an object being struck by a hammer. While in reality there is some compression of the hammer and of the object, and a small but finite time span during which the interaction occurs, that is not the way we normally see it. To the unaided eye, the whole thing takes place: Bang! – in an instant. This idealization not only corresponds to human intuition, but is very useful in physical applications.

Here an aside. In this discussion, when we use the term ‘physical’, we really mean ‘phenomenological’ – i.e. pertaining to the phenomena of nature. Thus, in our usage, the term physical could just as well apply to a problem in mathematical economics as to a problem in mechanics.

This still raises the question: Why create a whole theory to deal with an idea as simple as the Dirac Delta Function? Well, firstly, the idea may not be quite so simple as it looks. More importantly, the idea has important generalizations, each of which could be treated directly on its own merits, but only at the expense of an ever widening loss of clarity and comprehension.