So my next question is again about uniform continuity. Can you give me hints, or (better) give the solution of the following exercise? Thank you very much :-)
Given two subsets $A$ and $B$ of $\mathbb R$ with $A$ bounded from above (i.e., having an upper bound) and $B$ bounded from below (i.e., having a lower bound), where $\sup A = \inf B$ and $\sup A \in A\cap B$
(1) Prove that $A\cap B = \{\sup A\}.$
Now, take $A$ and $B$ as above. Let $f : \mathbb R \rightarrow \mathbb R$ and assume that $f$ is uniformly continuous on $A$ and on $B$.
(2) Prove that f is uniformly continuous on $A \cup B.$
My try:
(1) Let $x \in A\cap B$. Because $\sup A = \inf B,$ $\inf B \le x \le \sup A$ implies $x = \sup A$. I chose $x$ arbitrary, so $ A\cap B = \sup A$
(2) ???