Give an example of two open sets A, B and a continuous function $f:A\cup B\rightarrow\mathbb{R}$ such that $f|A$ and $f|B$ are uniformly continuous but $f$ is not.
I have been stuck in this one for a while. I just don't seem to be able to grasp how that function would exist. Maybe it has something to do with x's and y's at the edge of each set which satisfy $|x-y|<\delta$ but not $|f(x)-f(y)|<\epsilon$? But f has to be continuous!
Some hints to guide me on the right track would be appreciated.