What is the value of $f(100)$? 
We have $f:\Bbb R\to \Bbb R^*$, a function that admits primitives and admits the relations $$\cos \left(f(x)\right)=1,\ ∀x\in \Bbb R, \quad\text{and}\quad|f(\pi )−\pi |≤\pi .$$
What  is the value of $f(100)$?

My thought. We obviously have
$$\cos (f(100)) =1\overset{?}{\implies} f(100) =\arccos (1),$$
but this seems not to make any sense at all.
How can I use the provided inequality $|f(\pi)−\pi|≤\pi$?

Clearly $f(x)=2\pi k_x$ where $k_x\in\mathbb Z$ Besides
$$|2\pi k_x-\pi|\le\pi\iff-\pi\le2\pi k_x-\pi\le\pi\Rightarrow 0\le k_x\le1$$
Since $k_x$ is an integer the problem gives two solutions $f(x)=0$ and $f(x)=2\pi$ but because of the function is from $\Bbb R$ to $\Bbb R^*$ the only solution is
$$f(x)=2\pi$$
