Real Analysis: Uniform continuity proof

**tonyc4l** · November 19th 2009, 10:48 PM

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

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

**Drexel28** · November 19th 2009, 11:06 PM

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

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

What have you tried? You need two things that don't "react" well with one another.

**tonyc4l** · November 19th 2009, 11:25 PM

Originally Posted by Drexel28

What have you tried? You need two things that don't "react" well with one another.

Well, I saw in a subsequent problem that given the same setup, if you add the condition that each $X_i$ is compact, then you get that $f$ must be uniformly continuous on $X$ . So I'm trying to think of what would happen in this problem if one or both of $X_1$ or $X_2$ was not closed or unbounded. I also tried thinking of a counterexample, but I wasn't sure of how general I should be. Could a counterexample just be a specific $f, X_1$ , and $X_2$ ?

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