Find out where function is unfiormly continuous

Question

Let $f(x)=\sin(\ln x)$ for $x>0$

find such $a,b,c,d>0$ where $f(x)$ is:

uniformly continuous at intervals $(0,a], [b,+\infty)$

and Lipschitz at $(0,c]$ , $[d,+\infty)$

Answer 1

Hint: Check Mean value theorem.

Answer 2

Hint: If f is differentiable on an open subset $U$, then: $$ f' \text{ is bounded on } U \Leftrightarrow f \text{ is Lipschitz on } U$$