Definition:Supremum Metric/Differentiability Class

Definition

Let $\closedint a b \subseteq \R$ be a closed real interval.

Let $r \in \N$ be a natural number.

Let $\mathscr D^r \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$ which are of differentiability class $r$.

Let $d: \mathscr D^r \closedint a b \times \mathscr D^r \closedint a b \to \R$ be the function defined as:

$\ds \forall f, g \in \mathscr D^r \closedint a b: \map d {f, g} := \sup_{\substack {x \mathop \in \closedint a b \\ i \in \set {0, 1, 2, \ldots, r} } } \size {\map {f^{\paren i} } x - \map {g^{\paren i} } x}$

where:

$f^{\paren i}$ denotes the $i$th derivative of $f$
$f^{\paren 0}$ denotes $f$
$\sup$ denotes the supremum.

$d$ is known as the supremum metric on $\mathscr D^r \closedint a b$.

Also known as

This metric is also known as the sup metric or the uniform metric.

Examples

Difference between $C^0$ and $C^1$ Supremum Metrics

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:

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