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**Math2010** Prove or disprove (by counterex.) $\displaystyle f(x) = xlog(\frac{1}{x})$ is unif. cont. on (0,1) and prove or disprove $\displaystyle f(x) = \frac{cosx}{mx+b}$ is unif. cont. on (0,1) for all nonzer0 $\displaystyle m,b \in \mathbb{R}$

So I tried getting both into Lipschitz but was not successful. So all I know to do is for the first one is say: Given $\displaystyle \epsilon > 0$ set $\displaystyle \delta = ?$. If $\displaystyle x,a \in (0,1)$, then $\displaystyle |xlog\frac{1}{x}+alog\frac{1}{a}|\le |xlog\frac{1}{x}| + |alog\frac{1}{a}| < 2 $ (it's always going to be < 2 because our interval is (0,1)). Then I say if $\displaystyle (x,a) \in (0,1)$ and $\displaystyle |x-a| < \delta$ then $\displaystyle |f(x) - f(a)| = |xlog\frac{1}{x}-alog\frac{1}{a}|$ .... How do I know what I should set delta to be? Also I am totally lost starting here. Let me know if these is even remotely on the right track because I don't know for sure. Thank you so much.