How to prove this function is not uniformly continuous?

Question

I need to determine whether this function $f(x)=\log(2+\cos(e^x))$ is uniformly continuous on $\mathbb{R}$. I know this function is not uniformly continuous already from the graph of it, but I have no idea how to prove it formally. And I tried to take the derivative of this function and show that it is not bounded on $\mathbb{R}$. But I don't know how to prove that formally. Any suggestions appreciated.

Answer 1

Let $x_n=\log(2n\pi+\frac {\pi} 2)$ and $y_n=\log(2n\pi+\pi)$. Then, by MVT, $|x_n-y_n|\leq (\frac {\pi} 2) \frac 1 {t_n}$ for some $t_n$ between $2n\pi+\frac {\pi} 2$ and $2n\pi+\pi$. Thus $x_n -y_n \to 0$ as $ n \to \infty$. However $f(x_n)-f(y_n)\to \log 2$. Hence $f$ is not uniformly continuous.