I'm misunderstanding the theorem. Take f(x) = x + 1. That is a continuous function. But there does not exist an x where f(x) ~ x holds. What am I missing?
That's because there is also a precondition for the domain to be bounded, and since the real numbers are not bounded, the theorem does not apply here (or for any function R -> R).
You can just do the same thing in the integers modulo any value greater than 1, creating a bounded domain without creating a fixed point. I imagine this runs afoul of a different precondition which I haven't identified, though -- it's obviously way more likely than that the entire field has managed to miss something this obvious.
Edit: Actually it's obviously just that this set is not convex, or even continuous.
It must be a compact convex set. The entire reals is not compact.