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

Example of Supremum Metric on Differentiability Class

Let $f$ and $g$ be elements of $\mathscr D^1 \closedint 0 1$ defined as:

$\forall x \in \closedint 0 1: \begin {cases} \map f x = 0 \\ \map g x = 10^{-6} \map \sin {10^{16} x} \end {cases}$

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 $\pounds 30, 000$.

We have that:

$\ds \map {\max_{x \mathop \in \closedint 0 1} } g$

$=$

$\ds 10^{-6}$

$\ds \map {\min_{x \mathop \in \closedint 0 1} } g$

$=$

$\ds -10^{-6}$

and so the bumps are $\dfrac {1 \ \mathrm {km} } {10^6} = 1 \ \mathrm {mm}$ high.

Let $I$ be the distance along the road according to $g$.

We have that:

$\ds I$

$=$

$\ds \int_0^1 \sqrt {1 + \paren {\map {g'} x}^2 } \rd x$

Definition of Arc Length

$\ds $

$=$

$\ds \int_0^1 \sqrt {1 + 10^{20} \map {\cos^2} {10^{16} x} } \rd x$

$\ds $

$\ge$

$\ds \int_0^1 10^{10} \size \cos {10^{16} x} \rd x$

We have that $\size {\cos 10^{16} x} \ge \dfrac 1 2$ for $\dfrac 2 3$ of the range $\closedint 0 1$.

So:

$I \ge \dfrac {10^{10} } 3$

which costs more than $3 \times 10^6 \ \mathrm p = \pounds 30, 000$.

$\blacksquare$

1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 27$