I had an analysis exam today, and the last question was: Give an example of a set F\subset\mathcal{C}_b(\mathbb{R},\mathbb{R}) that is pointwise compact, (uniformly) equicontinuous, and NOT totally bounded.

I know F has to be open, otherwise by the Arzela-Ascoli theorem, it would be compact and therefore totally bounded, but unfortunately, I got to this question with ~3 minutes left, and was only able to come up with F=\{n\sin(x/n)~|~n\in\mathbb{N}\}. I am pretty sure this doesn't work, so what would an example of a function with the above properties be?

Note that \mathcal{C}_b(\mathbb{R},\mathbb{R}) is the space of all bounded functions from the reals to the reals.