Continuous functions on $(0,1)$ that have no fixed point

Question

I need an example of a continuous function that does not have a fixed point, a function $f:(0,1)\to(0,1)$ with domain $(0,1)$ and image $(0,1)$. Thanks.

Answer 1

$$ f(x)=x^2 $$ Also, $f(x)=x^a$, for every $a\ne 1$, $a>0$.

Answer 2

Any continous function will do as long as its graph does not intersect the diagonal line $y=x$. So for example the function $$x\mapsto \frac x2$$ will work, since $ \frac x2<x$ in the region of interest. Similarly $$x\mapsto\frac {x+1}2$$ or $$ x\mapsto \min\bigg(\frac x3, \frac14\bigg)$$ or what have you.