Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 12G / a / iii

The closed interval does have the fixed-point property. If $f(x,0)=(F(x),0)$, then $F(0)-0\ge 0$ and $F(1)-1\le 0$. The intermediate value theorem supplies $x\in [0,1]$ with $F(x)-x=0$.