Real Analysis: Uniform continuity proof

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 is uniformly continuous on each of the sets and let . Show that need not be continuous on . Show that, even if is continuous on , need not be uniformly continuous on .

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