An equation that is undefined at every value except one (plotting a single point)

Question

Suppose I wanted to plot an equation that would only place a single point at a specific (x,y) value, and nothing anywhere else, that is, the equation, y=g(x), would be undefined for all other x values except the x from that one (x,y) point. What might that look like?

I toyed around with dividing by zero in the equation (to make all x-values undefined and therefor not plotted), but I wasn’t sure how to not divide by zero IFF x is the x-value I want plotted. I know Kronecker Delta function could be used here but afaik it only has a definition as a piecewise function and obviously that defeats the entire purpose of what I’m trying to do. I suspect the answer will involve limits. Any ideas?

I also have an idea of plotting the equation for a circle with its center at the (x,y) point and the radius approaching zero, but I don’t know enough about limits to know if that will suffice?

Answer 1

In the context of real functions $$f(x) = \sqrt x + \sqrt{-x}$$

has a graph consisting of only the point $(0,0)$

Answer 2

The graph of the function f = {(x,y)} is a single point.
The constant function f:{0} -> {1}, x -> 1, for example.

You could conjure up a meaningless monster like
f(x) = x/0 if x /= 0; f(0) = 0/0 = 1 or whatever.

I don't see any slick answer for a dubious problem.
Use the set theory definition of function
instead of a naive calculus notion.

Answer 3

You can just write $(x-x_0)^2+(y-y_0)^2=0$. This equation has only the point $(x_0,y_0)$ as solution.