real analysis - Fixed point and period of continuous function

Prove/ Disprove:

1. Let $f:(0,1)\to(0,1)$ be such that $|f(x)-f(y)|\leq 0.5|x-y|$ for all $x ,y.$ Then f has a fixed point.

   
   
   

   2.Let $f:\mathbb R\to\mathbb R$ be continuous and periodic with period $T>0.$Then there exists a point $x_0\in\mathbb R$ such that

   
   
   
   

   $f(x_0)=f(x_0+T/2).$

Answer

• Let $f(x)=\frac{x}{2}$ so $f$ hasn't a fixed point in $(0,1)$.




• Let $g(x)=f(x+T/2)-f(x)$ then $g$ is continuous and $g(0)g(T/2)\le0$ so use the intermediate value theorem to conclude.