How to plot a generalized function?

When we are dealing with regular functions, one thing we can do with them is to plot them. We would draw $y$ and $x$ coordinate axes, draw some line and would say that the line represents some function $y(x)$.

When we are dealing with distributions, we call them generalized functions. If we call them functions, there ought to be a way to draw them. But I can't figure out how to do that.

Consider, for instance, a Dirac delta distribution of a test function $\phi$: $$\delta (x-a) [\phi] \equiv \phi(a)$$ It matches the function $\phi$ with its value at $x=a$. Therefore, in order to plot it I have to assign the vertical axis to $\phi(a)$ (ordinates) and the horizontal axis to $\phi$ (abscissas).

Since $\phi(a)$ is scalar I understand perfectly well how to put its values on the vertical axis.

But for the life of me I can't understand what it means to put function $\phi$ on the horizontal axis.

I'm wondering how mathematicians plot generalized functions and if they don't then what the usual discussion about this peculiarity is.

Solution 1:

First, for just Dirac deltas at various locations, I graph them by drawing an arrow up or down at the location, maybe labelled by a constant coefficient. This has helped me, and seems to help some other people, understand that the derivative of Heaviside's step function is Dirac delta (at 0) without having to integrate by parts.

Also, and for more complicated distributions, I do find it useful to draw classical-function approximations. This can be useful to understand derivative-taking and such (since, after all, the weak-dual limit of distributional/classical derivatives is the distributional derivative of the weak-dual limit).