Supremum Metric on Differentiability Class/Examples/Difference between C0 and C1

Example of Supremum Metric on Differentiability Class
Let $\mathscr D^1 \closedint 0 1$ be the set of all continuous functions $\phi: \closedint 0 1 \to \R$ which are of differentiability class $1$.

Let $f$ and $g$ be elements of $\mathscr D^1 \closedint 0 1$ defined as:
 * $\forall x \in \closedint 0 1: \begin {cases} \map f x = 0 \\ \map g x = 10^{-6} \map \sin {10^{16} x} \end {cases}$

Let $d_0$ denote the supremum metric $C^0$ on $\mathscr D^1 \closedint 0 1$:
 * $\ds \forall f, g \in A: \map {d_0} {f, g} := \sup_{x \mathop \in \closedint 0 1} \size {\map f x - \map g x}$

Let $d_1$ denote the supremum metric $C^1$ on $\mathscr D^1 \closedint 0 1$:
 * $\ds \forall f, g \in A: \map {d_1} {f, g} := \sup_{x \mathop \in \closedint 0 1} \size {\map f' x - \map g' x}$

Then:
 * $\map {d_0} {f, g} = 10^{-6}$

while:
 * $\map {d_1} {f', g'} = 10^{10}$

Proof
We have that:

Then we have that: