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Show $f:(0,\infty)\rightarrow \mathbb{R}, f(x)=x\ln x$ on $(0,\infty)$ is not uniformly continuous.

I think that the general way to prove that something is not continuous in a metric space is to let $\epsilon=...$ and show that $\forall\delta>0$, $d'(f(x)-f(y))>\epsilon$. I can't use the Mean Value Theorem because we haven't gone over it yet.

Here's an attempt:
Let $\epsilon=1$. Without loss of generality, let $x>y>0$. $|x\ln x-y\ln y|<|(x-y)(\ln x)|\leq|x-y||\ln x|$...$\delta=\frac{\epsilon}{\ln x}$ and therefore doesn't work for all $x$?

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