Continuous real function/Crossing/Fixed point/Exercise

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be a continuous function, and suppose that there exist ${}x,y\in \mathbb {R}$ with

${}f(x)\leq x\,$

and

${}f(y)\geq y\,.$

Show that ${}f$ has a fixed point.

Create a solution