1. ## help

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

$f_1(x) = x$ is uniformly continuous

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

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

3. 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.)

4. Originally Posted by Krizalid
$h(x)=\sin x$ is also uniformly continuous. (Actually Lipschitz.)
Don't you mean $h(x)=x \sin x$

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

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