If $~f~$ is continuous and $f(0)=\frac{\pi}{2},~$ $g(x)=\cos(f(x))$ is monotonically decreasing, then $~f~$ is uniformly continuous

Question

Let $~f,~g : \Bbb R \to \Bbb R~$, then show that if $f$ is continuous and $~f(0)=\frac{\pi}{2},~~g(x)=\cos(f(x))~$ is monotonically decreasing, then $~f~$ is uniformly continuous.

My thought:

I think we need to show that $f$ can be continuously extended. So, I tried to suppose $\lim_{x\to \infty}f(x)$ dose not exist and induce a contradiction to the fact that $\cos(f(x))$ is monotone decreasing function... Could you please any hint to solve this problem by myself ?

I don't want you to show all proof.

Thank you.