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.