# Banach fixed-point theorem and  the map $Tf(x)=\int_0^x f(s)ds$ on $C[0,1]$

Let $T: C[0,1] \rightarrow C[0,1]$ be a map defined by
$Tf(x)=\int_0^x f(s)ds.$
Prove that
i)  $T$ is not a contraction.
ii) $T$ has a unique fixed point.
iii) $T^2$ is a contraction.