# Uniform continuity on union of sets

If a function $f(x)$ is uniformly continuous on two completely separated intervals $[a,b]$ and $[c,d]$, then is it true that $f(x)$ is uniformly continuous on $[a,b]\cup [c,d]$?

I also think that $[a,b]$ and $[c,d]$ are compact sets, so $[a,b]\cup [c,d]$ is also a compact set. Hence $f(x)$ is uniformly continuous on $[a,b]\cup [c,d]$. I'm not sure. Am I right?

• you are right. It follows more or less directly from the definition. no need going through compactness. – user251257 Aug 20 '15 at 10:31
• thank you very much, sir.... – DEEP Aug 20 '15 at 15:03