# Multiplying Uniformly Continuous functions

• Mar 20th 2010, 08:25 PM
CrazyCat87
Multiplying Uniformly Continuous functions
If f and g are two uniformly continuous functions, I need to show that their product is also uniformly continuous. I started with the definition of uniform continuity but got nowhere...
• Mar 20th 2010, 08:58 PM
Tinyboss
You're not going to get anywhere, either, unless you restrict the domain. Is $x\mapsto x$ uniformly continuous on $\mathbb{R}$? What about $x\mapsto x^2$?
• Mar 21st 2010, 04:28 PM
CrazyCat87
Sorry I'm not sure what you mean by that...
• Mar 21st 2010, 04:30 PM
Tinyboss
Let $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=x$. Is f uniformly continuous? What about the product of f with f?
• Mar 23rd 2010, 05:45 PM
CrazyCat87
yea I haven't figured it out yet... it's probably simple but i'm having a lot of trouble with this one
• Mar 23rd 2010, 07:04 PM
Krizalid
$f(x)=x$ is trivially uniformly continuous since $\left| f(x)-f(y) \right|=\left| x-y \right|,$ so the product is not uniformly continuous, namely $f(x)f(x)=x^2$ is not uniformly continuous.
• Mar 23rd 2010, 08:00 PM
CrazyCat87
lol but this is isn't making sense since I'm supposed to show the product IS uniformly continuous
• Mar 24th 2010, 07:32 AM
Krizalid
IT'S NOT TRUE, and i've provided a counterexample.
• Mar 24th 2010, 02:09 PM
CrazyCat87
lol you're right, now what if both functions were bounded?
• Mar 25th 2010, 07:43 PM
phuongtim39
Quote:

Originally Posted by CrazyCat87
yea I haven't figured it out yet... it's probably simple but i'm having a lot of trouble with this one

I accept with information: http://www.mathhelpforum.com/math-he...300389f6-1.gif is trivially uniformly continuous since http://www.mathhelpforum.com/math-he...48476b03-1.gif
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