Continuous function on complete bounded metric space need not be bounded I came across the following old qual problem:

Suppose $(X,d)$ is a complete metric space with finite diameter.  Is every continuous function on $X$ bounded?

It seems like the function $1/x$ on the interval $(0,1)$ with the discrete metric is a counterexample.  Is it this simple, or am I missing something?

Yes, your example is valid. Another example which captures the  same essence as yours a bit more compactly would be the identity function on $\mathbb N$ with the discrete metric. (Considering the range of the map to be $\mathbb N$ with the usual metric)
