Let $S$ denote the set of continuous real-valued functions on the interval $[0,1]$. For $f, g \in S$, set

$$d_{1}(f, g)=\sup \{|f(x)-g(x)|: x \in[0,1]\} \quad \text { and } \quad d_{2}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x$$

Show that both $d_{1}$ and $d_{2}$ define metrics on $S$. Does the identity map on $S$ define a continuous map of metric spaces $\left(S, d_{1}\right) \rightarrow\left(S, d_{2}\right) ?$ Does the identity map define a continuous map of metric spaces $\left(S, d_{2}\right) \rightarrow\left(S, d_{1}\right)$ ?