Define a contraction mapping and state the contraction mapping theorem.

Let $C[0,1]$ be the space of continuous real-valued functions on $[0,1]$ endowed with the uniform norm. Show that the map $A: C[0,1] \rightarrow C[0,1]$ defined by

$$A f(x)=\int_{0}^{x} f(t) d t$$

is not a contraction mapping, but that $A \circ A$ is.