# uuniform continuety question

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• Sep 3rd 2011, 01:11 PM
transgalactic
uuniform continuety question
4)b)
prove that $f(x)=sin(lnx)$continues uniformly in $[1,\infty)$
4)c)
prove that for all $\delta>0$exists natural 'n' so $e^{\frac{\pi}{2}-2\pi n}-e^{-2\pi n}<\delta$
4)d)
prove that $f(x)=sin(lnx)$ is not uniformly continues in (0,1)
in part d we can use part c

regarding part A:
i want to show that the derivative of f(x) is bounded then its automatickly uniformly continues.
$f'(x)=\frac{cos(lnx)}{x}\leq \frac{1}{x}$
so when we got to infinity the 1/x goes to zero.but if we got to x=0 the 1/x goes to infinity
why it is bounded
?