Prove that intervals of the form $(a,b]$, $[a,b)$, $(-\infty,a]$, $[a,\infty)$ do not have the fixed point property.

Question

Prove that intervals of the form $(a,b]$, $[a,b)$, $(-\infty,a]$, $[a,\infty)$ do not have the fixed point property.

In the case of open intervals, I can derive that they do not have the fixed point property by the fact that the real line doesn't have this property. But how can I show this in these kinds of half intervals?

Answer 1

Let's take one of them for example: the first one, $(a,b]$.

Say, for simplicity, that $a=0$ and $b=1$. Now take the function $x\mapsto \frac x2$. It is continuous, from $(a,b]$ to $(a,b]$, but does not have fixed points.