jump to navigation

Uniformly continuous function on a bounded set March 3, 2011

Posted by Phi. Isett in Calculus.

Here’s a problem which is a little odd and kind of interesting; I’ll write it down before I forget about it.

Every function f : (a,b) \to {\mathbb R} which is uniformly continuous on a bounded interval is bounded.  You can prove this by bounding f(x) – f(y) by breaking the total change into a bunch of small changes (using uniform continuity) just as in one common proof of the Fundamental Theorem of Calculus (this technique also gets used in the proof of Sard’s theorem, Harnack’s inequality, and some other things).

Now replace the bounded interval (a,b) by a bounded subset S \subseteq {\mathbb R}.  Exercise: are uniformly continuous functions still bounded?



No comments yet — be the first.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: