uuniform continuety question

4)b)

prove that $\displaystyle f(x)=sin(lnx)$continues uniformly in $\displaystyle [1,\infty)$

4)c)

prove that for all $\displaystyle \delta>0 $exists natural 'n' so $\displaystyle e^{\frac{\pi}{2}-2\pi n}-e^{-2\pi n}<\delta$

4)d)

prove that $\displaystyle f(x)=sin(lnx)$ 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.

$\displaystyle 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

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