# Thread: Real Analysis: Uniform continuity proof

1. ## 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 $\displaystyle f$ 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.

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

3. 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 $\displaystyle X_i$ is compact, then you get that $\displaystyle f$ must be uniformly continuous on $\displaystyle X$. So I'm trying to think of what would happen in this problem if one or both of $\displaystyle X_1$ or $\displaystyle X_2$ 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 $\displaystyle f, X_1$, and $\displaystyle X_2$?