Prove that an Interval I is closed and bounded.

Question

I'm attempting to prove the following statement:

Let $I$ be a non-trivial interval over the real numbers. Show that if every continuous function on $I$ is uniformly continuous, then $I$ is closed and bounded.

Answer 1

If $I$ is unbounded, for example $I = (a, \infty)$, then $f(x) = x^2$ is uniformly continuous -- a contradiction.

If $I$ is not closed , for example $I = (a, b) $, then $f(x) = 1/(x-a)$ is uniformly continuous -- a contradiction.