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

From ProofWiki
Jump to navigation Jump to search

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:

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


Application

Consider a road along a $1 \ \mathrm {km}$ route whose surface can be described as $\map g x$.

Let a journey be made along this road be charged at a rate of $1$ penny per $1000 \ \mathrm {km}$.

Suppose the distance over which this rate is measured is taken along the road surface.

Then, although the bumps are $1 \ \mathrm {mm}$ high, the cost of the journey is over $\pounds 30, 000$.


Proof

We have that:

\(\ds \map {\max_{x \mathop \in \closedint 0 1} } g\) \(=\) \(\ds 10^{-6}\)
\(\ds \map {\min_{x \mathop \in \closedint 0 1} } g\) \(=\) \(\ds -10^{-6}\)
\(\ds \leadsto \ \ \) \(\ds \sup_{x \mathop \in \closedint 0 1} \size {\map f x - \map g x}\) \(=\) \(\ds 10^{-6}\)
\(\ds \leadsto \ \ \) \(\ds \map {d_0} {f, g}\) \(=\) \(\ds 10^{-6}\)

Then we have that:

\(\ds \dfrac \d {\d x} \map g x\) \(=\) \(\ds 10^{16} \times 10^{-6} \map \cos {10^{16} x}\)
\(\ds \) \(=\) \(\ds 10^{10} \map \cos {10^{16} x}\)
\(\ds \leadsto \ \ \) \(\ds \map {\max_{x \mathop \in \closedint 0 1} } g'\) \(=\) \(\ds 10^{10}\)
\(\ds \map {\min_{x \mathop \in \closedint 0 1} } g'\) \(=\) \(\ds -10^{10}\)
\(\ds \leadsto \ \ \) \(\ds \sup_{x \mathop \in \closedint 0 1} \size {\map f' x - \map g' x}\) \(=\) \(\ds 10^{10}\)
\(\ds \leadsto \ \ \) \(\ds \map {d_1} {f, g}\) \(=\) \(\ds 10^{10}\)

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Supremum_Metric_on_Differentiability_Class/Examples/Difference_between_C0_and_C1&oldid=505349"