# help

• Jul 13th 2009, 10:12 AM
Kat-M
help
Is f(x)=x*sin(x) uniformly continuous? justify.

• Jul 13th 2009, 05:03 PM
pickslides

$\displaystyle f_1(x) = x$ is uniformly continuous

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

therefore $\displaystyle f(x) = x\times sin(x)$ is also uniformly continuous
• Jul 13th 2009, 05:42 PM
Krizalid
You're wrong.

Product of two uniformly continuous functions is not uniformly continuous.

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

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$\displaystyle h(x)=\sin x$ is also uniformly continuous. (Actually Lipschitz.)
• Jul 13th 2009, 05:53 PM
Jose27
Quote:

Originally Posted by Krizalid
$\displaystyle h(x)=\sin x$ is also uniformly continuous. (Actually Lipschitz.)

Don't you mean $\displaystyle h(x)=x \sin x$
• Jul 13th 2009, 05:54 PM
Krizalid
No, I was giving another example about another uniformly continuous function.
• Jul 13th 2009, 07:09 PM
pickslides
Good point Krizalid, also if you graph the function you can see it is not uniformly continuous.