Real Analysis: Uniform continuity proof

• Nov 19th 2009, 10:48 PM
tonyc4l
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.
• Nov 19th 2009, 11:06 PM
Drexel28
Quote:

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.
• Nov 19th 2009, 11:25 PM
tonyc4l
Quote:

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\$?