help

**Kat-M** · July 13th 2009, 10:12 AM

Is f(x)=x*sin(x) uniformly continuous? justify.

please help.

**pickslides** · July 13th 2009, 05:03 PM

Here's my adhoc attempt, please trump in anybody...

$f_1(x) = x$ is uniformly continuous

$f_2(x) = sin(x)$ is uniformly continuous

therefore $f(x) = x\times sin(x)$ is also uniformly continuous

**Krizalid** · July 13th 2009, 05:42 PM

You're wrong.

Product of two uniformly continuous functions is not uniformly continuous.

Cheap example: $f(x)=x$ is uniformly continuous and $g(x)=f(x)\cdot f(x)=x^2$ is not uniformly continuous.

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$h(x)=\sin x$ is also uniformly continuous. (Actually Lipschitz.)

**Jose27** · July 13th 2009, 05:53 PM

Originally Posted by Krizalid

$h(x)=\sin x$ is also uniformly continuous. (Actually Lipschitz.)

Don't you mean $h(x)=x \sin x$

**Krizalid** · July 13th 2009, 05:54 PM

No, I was giving another example about another uniformly continuous function.

**pickslides** · July 13th 2009, 07:09 PM

Good point Krizalid, also if you graph the function you can see it is not uniformly continuous.

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