Open Balls of Supremum Metric on Continuous Real Functions on Closed Interval

Theorem
Let $\mathscr C \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$.

Let $d: \mathscr C^2 \to \R$ be the supremum metric on $\mathscr C \closedint a b$ defined as:
 * $\ds \forall f, g \in \mathscr C \closedint a b: \map d {f, g} := \sup_{x \mathop \in \closedint a b} \size {\map f x - \map g x}$

where $\sup$ denotes the supremum.

Let $f, g \in \mathscr C \closedint a b$ be such that:
 * $\forall x \in \closedint a b: \map f x < \map g x$

Consider the set $S$, defined as:
 * $S = \set {h \in \mathscr C \closedint a b: \forall x \in \closedint a b: \map f x < \map h x < \map g x}$

Then:
 * $S$ is an open ball of $\struct {C \closedint a b, d}$


 * $\exists C \in \R_{>0}: \map g x := \map f x + C$
 * $\exists C \in \R_{>0}: \map g x := \map f x + C$

That is, $f$ and $g$ differ by a constant.

Proof
Recall the definition of open ball:

The open $\epsilon$-ball of $a$ in $M = \struct {A, d}$ is defined as:


 * $\map {B_\epsilon} a := \set {x \in A: \map d {x, a} < \epsilon}$

In this context, the open $\epsilon$-ball of $\phi$ in $C \closedint a b$ is defined as:


 * $\map {B_\epsilon} \phi := \set {\rho \in C \closedint a b: \sup_{x \mathop \in \closedint a b} \size {\map \rho x - \map \phi x} < \epsilon}$

Necessary Condition
Suppose $g \in \mathscr C \closedint a b$ is defined as:
 * $\forall x \in \closedint a b: \map g x := \map f x + C$

Let $h \in S$.

Sufficient Condition
Let $S$, defined as:
 * $S = \set {h \in \mathscr C \closedint a b: \forall x \in \closedint a b: \map f x < \map h x < \map g x}$

be an open ball in $\mathscr C \closedint a b$.

Thus there exists $\epsilon \in \R_{>0}$ and $\rho \in \mathscr C \closedint a b$ such that:
 * $\ds \exists \epsilon \in \R_{>0}: \forall \phi \in S: \sup_{x \mathop \in \closedint a b} \size {\map \rho x - \map \phi x} < \epsilon$

it is not the case $f$ and $g$ are such that:
 * $\forall x \in \closedint a b: \map g x - \map f x = C$

for some constant $C \in \R$.

Then:
 * $\exists \xi, \zeta \in \closedint a b: \map g \xi - \map f \xi \ne \map g \zeta - \map f \zeta$

, suppose $\map g \xi - \map f \xi > \map g \zeta - \map f \zeta$.

Then $\sup_{x \mathop \in \closedint a b} \size {\map g x - \map f x} \ge \map g \xi - \map f \xi > \map g \zeta - \map f \zeta$

As $f$ and $g$ are continuous:
 * $\exists \openint p q \subset \closedint a b: \forall x \in \openint p q: \size {\map g x - \map f x} > \map g \zeta - \map f \zeta$

Let $h \in S$ such that:
 * $\exists r, s \in \closedint a b: \size {\map h r - \map h s} > \map g \zeta - \map f \zeta$