Feather

That's not entirely true, Jesus. The Dirac Delta function, for example, has just the property you describe cannot exist (finite area with zero breadth). It's a fundamental part of much of basic physics theory, too (e.g. the charge density of a point charge, a very basic model for the surface periodic potential of solids in Quantum Mechanics, "unit basis vector operators" in certain vector spaces, and much more).

Friar Bellows

Nice catch, Feather. But somewhat in defence of Jesus Christ (funny, I never thought I would say that), the "Dirac delta function" is technically not a function, but a "linear functional", a sort of generalised function. Here, this explains it better than I ever could:

<a href="http://mathworld.wolfram.com/DeltaFunction.html" target="_blank">Delta Function</a>

Although, I still don't fully understand it.

wade-w

A linear functional is a special case of a linear transformation, in which the codomain is the real numbers. In the case of the delta function, the domain is a set of functions, which is where the problem comes in. We are mapping a set of functions to the reals. I'm not sure that talking about zero breadth is meaningful in this context.

Its still a function. Its domain just doesn't happen to be a subset of the real numbers.

Feather

Yeah, I knew that, Friar, but I figured it might irk some mathematicians to put it the way I did. You know, kinda like integrating F(x)dx from 0 to x.