Is $f(x)=\ln(2+\sin(x))$ uniformly continuous on $\mathbb{R}$?

Question

I have to prove that $f(x)=\ln(2+\sin(x))$ is uniformly continuous without the mean value theorem! How can I get a good procedure?

Answer 1

Show that any periodic continuous function is uniformly continuous. (Use that continuous with compact domain is uniformly continous)

Answer 2

Hinr: The function is continuous on the closed interval $[0,2\pi]$ and periodic.

Answer 3

You could also use the mean value theorem. Verify that $|f'|\le 1$ everywhere, which shows by the MVT that $|f(y)-f(x)|\le |y-x|$ for all $x,y.$