We all know that the ring of germs of continuous functions at a point on, say $\mathbb{R}$, has a unique maximal ideal- namely, those functions that vanish at that point.

**Can anyone think of a single other example of a prime ideal?**

We know that there have to be lots and lots of them, since there are no non-zero nilpotent elements in the ring. You can't try and construct such a thing by looking at the germ of functions that vanish at some bigger set of points, because then you can use bump functions to produce two germs whose product is in the ideal but neither of whom belong to the ideal themselves.

The only other possible things we could think of would only have a hope of working if we were dealing with smooth functions (though we still haven't come up with an example there!).

Is it even possible to write down an example of such a thing for continuous functions, or are they just too ugly?