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...

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- Mar 20th 2010, 07:25 PMCrazyCat87Multiplying 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, 07:58 PMTinyboss
You're not going to get anywhere, either, unless you restrict the domain. Is $\displaystyle x\mapsto x$ uniformly continuous on $\displaystyle \mathbb{R}$? What about $\displaystyle x\mapsto x^2$?

- Mar 21st 2010, 03:28 PMCrazyCat87
Sorry I'm not sure what you mean by that...

- Mar 21st 2010, 03:30 PMTinyboss
Let $\displaystyle f:\mathbb{R}\to\mathbb{R}$ be given by $\displaystyle f(x)=x$. Is f uniformly continuous? What about the product of f with f?

- Mar 23rd 2010, 04:45 PMCrazyCat87
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, 06:04 PMKrizalid
$\displaystyle f(x)=x$ is trivially uniformly continuous since $\displaystyle \left| f(x)-f(y) \right|=\left| x-y \right|,$ so the product is not uniformly continuous, namely $\displaystyle f(x)f(x)=x^2$ is not uniformly continuous.

- Mar 23rd 2010, 07:00 PMCrazyCat87
lol but this is isn't making sense since I'm supposed to show the product IS uniformly continuous

- Mar 24th 2010, 06:32 AMKrizalid
IT'S NOT TRUE, and i've provided a counterexample.

- Mar 24th 2010, 01:09 PMCrazyCat87
lol you're right, now what if both functions were bounded?

- Mar 25th 2010, 06:43 PMphuongtim39
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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