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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?

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

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