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**tonyc4l** I need some help/advice doing the following proof. Also, I'm in a beginning real analysis class and the section we're covering is on uniform continuity, so keep that in mind I guess.

Suppose $\displaystyle f$ is uniformly continuous on each of the sets $\displaystyle X_1, X_2, ..., X_n$ and let $\displaystyle X = \bigcup_{i=1}^{n} X_i$. Show that $\displaystyle f$ need not be continuous on $\displaystyle X$. Show that, even if $\displaystyle f$ is continuous on $\displaystyle X$, $\displaystyle f$ need not be uniformly continuous on $\displaystyle X$.

I think it should suffice to show the result for $\displaystyle X = X_1 \cup X_2$, since you could do a simple induction proof to show the result for up to $\displaystyle X_n$. Other than that though, I'm confused as to where I should even begin. Any help is greatly appreciated.