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Let $f:[0,\infty)\rightarrow \mathbb R$ be a function defined by

$f(x)=\large \large \{x^{2/3}log x,x>0$

                  0 x=0

Then is uniformly continuous on [0,$\infty$)

f is definitely continuous at x=0 as $lim_{x\rightarrow 0}$f(x)=0

f is uniformly cont. iff $lim_{x\rightarrow 0}$f(x) and $lim_{x\rightarrow \infty}$f(x) both exist.However not sure about the second limit .How to show it?

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Outline: Note that we have continuity on the closed interval $[0,1]$.

And on $[1,\infty)$ the derivative is bounded.

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  • $\begingroup$ You are welcome. $\endgroup$ – André Nicolas Nov 15 '14 at 6:39

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