Consider the metrics spaces $(X_1,d_\infty)$ and $(X_2,d_1)$ where $X_i=C[0,1]$ ,$d_\infty$ is the 'sup' metric and $d_1$ is the metric defined by $$d_1(f,g)=\Bigg(\int_0^1 \vert f-g \vert^2 \Bigg)^\frac{1}{2}$$
How to prove or disprove the identity function $f:(X_1,d_\infty) \rightarrow \ (X_1,d_1)$ is continuous ?
Can I have a hint?
Hint: prove that the distance in the sup metric is greater than or equal to the distance in the integral metric, then try to use that to your advantage.