Uniformly continuous on each of $A$ and $B$, not on $A\cup B$

Question

I have a homework problem that asks me to find a function defined on subsets $A$ and $B$ of $\mathbb R$ such that the function is uniformly continuous on each of $A$ and $B$, and is continuous but not uniformly continuous on $A \cup B$.

I have the example of the "jump-discontinuity" (where function is discontinuous at a point, period).

I also tried using that functions mapped from cpt sets to reals will be uniformly continuous...but I can't seem to figure it out.

I'd appreciate a hint, or anything! Thank you!

Answer 1

Let $A=[0,1)$, $B=(1,2]$, and let $f$ be constant on each of $A$ and $B$, with different values.