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. 


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Erdos Erdos
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