How to conclude that $f$ may not have a fixed point

Question

Let $f:[0,1)\rightarrow [0,1)$ be a continuous map.How to conclude that $f$ may /may not have a fixed point?

I tried to prove this fact considering the function $g(x)=f(x)-x $ then $g$ is continuous $g(0)=0$

But the proof does not hold as in the case for$[0,1]$

Find a counter e.g. say like $2-\sqrt 2^{1-x}$. – Macavity Feb 04 '15 at 10:52 — Macavity, Feb 04 '15 at 10:52

score 1 · Answer 1 · answered Feb 04 '15 at 10:41

1

The segment $[0,1)$ is homeomorphic to the ray $[0,\infty)$. If you find a function $f:[0,\infty)\to[0,\infty)$ with no fixed point, you can thus deduce the existence of such a function $[0,1)\to[0,1)$.

answered Feb 04 '15 at 10:41

Amitai Yuval

19,308

How to conclude that $f$ may not have a fixed point

1 Answers1