Let $C[0,1]$ be the set of continuous real-valued functions on $[0,1]$ with the uniform norm. Suppose $T: C[0,1] \rightarrow C[0,1]$ is defined by

$$T(f)(x)=\int_{0}^{x} f\left(t^{3}\right) d t$$

for all $x \in[0,1]$ and $f \in C[0,1]$. Is $T$ a contraction mapping? Does $T$ have a unique fixed point? Justify your answers.