Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 1 / 10E / iii

No. Define $f(x)=1$ for $0\le x<1$ and $f(1)=0$. This self-map of $[0,1]$ is discontinuous at one. Every point smaller than one is sent to a different point, and one is sent to zero, so it has no fixed point.