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?