Oscillation at Point (Infimum) equals Oscillation at Point (Limit)

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Theorem

Let $f: D \to \R$ be a real function where $D \subseteq \R$.

Let $x$ be a point in $D$.

Let $N_x$ be the set of neighborhoods of $x$.

Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$ as defined by:

$\map {\omega_f} x = \inf \set {\map {\omega_f} {I \cap D}: I \in N_x}$

where $\map {\omega_f} {I \cap D}$ is the oscillation of $f$ on a real set $I \cap D$:

$\map {\omega_f} {I \cap D} = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$

Let $\map {\omega^L_f} x$ be the oscillation of $f$ at $x$ as defined by:

$\map {\omega^L_f} x = \ds \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} \cap D}$

Then:

$\map {\omega_f} x \in \R$ if and only if $\map {\omega^L_f} x \in \R$

and, if $\map {\omega_f} x$ and $\map {\omega^L_f} x$ exist as real numbers:

$\map {\omega_f} x = \map {\omega^L_f} x$

Proof

Lemma

Let $f: D \to \R$ be a real function where $D \subseteq \R$.

Let $x$ be a point in $D$.

Let $N_x$ be the set of neighborhoods of $x$.

Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$ as defined by:

$\map {\omega_f} x = \ds \inf \set {\map {\omega_f} {I \cap D}: I \in N_x}$

where $\map {\omega_f} {I \cap D}$ is the oscillation of $f$ on a real set $I \cap D$:

$\map {\omega_f} {I \cap D} = \ds \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$

Let $\map {\omega^L_f} x$ be the oscillation of $f$ at $x$ as defined by:

$\map {\omega^L_f} x = \ds \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} \cap D}$

Let $\map {\omega^L_f} x \in \R$.

Let $\map {\omega_f} x \in \R$.

Then $\map {\omega^L_f} x = \map {\omega_f} x$.

$\Box$

Necessary Condition

Let $\map {\omega_f} x \in \R$.

We need to prove:

$\map {\omega^L_f} x \in \R$

$\map {\omega^L_f} x = \map {\omega_f} x$

where:

$\map {\omega^L_f} x = \ds \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} \cap D}$

$\map {\omega_f} {\openint {x - h} {x + h} \cap D} = \sup \set {\size {\map f y - \map f z}: y, z \in \openint {x - h} {x + h} \cap D}$

$\map {\omega_f} x = \inf \set {\map {\omega_f} {I \cap D}: I \in N_x}$

$\map {\omega_f} {I \cap D} = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$

Let $\epsilon \in \R_{>0}$.

Then an $I \in N_x$ exists such that:

$\map {\omega_f} {I \cap D} - \map {\omega_f} x < \epsilon$ by Infimum of Set of Oscillations on Set is Arbitrarily Close

Let $I$ be such an element of $N_x$.

We observe in particular that $\map {\omega_f} {I \cap D} \in \R$.

A neighborhood in $N_x$ contains an open subset that contains the point $x$.

So, $I$ contains such an open subset as $I \in N_x$.

Therefore, a $\delta \in \R_{>0}$ exists such that $\openint {x - \delta} {x + \delta}$ is a subset of $I$.

Let $h$ be a real number that satisfies: $0 < h < \delta$.

We observe that $\openint {x - h} {x + h} \subset I$.

Also, $\openint {x - h} {x + h} \in N_x$.

We have:

$I \in N_x$

$\openint {x - h} {x + h} \in N_x$

$\openint {x - h} {x + h} \subset I$

$\map {\omega_f} {I \cap D} \in \R$

from which follows by Oscillation on Subset:

$\map {\omega_f} {\openint {x - h} {x + h} \cap D} \in \R$

$\map {\omega_f} {\openint {x - h} {x + h} \cap D} \le \map {\omega_f} {I \cap D}$

We have that:

$\map {\omega_f} {\openint {x - h} {x + h} \cap D} \in \set {\map {\omega_f} {I' \cap D}: I' \in N_x}$

as $\openint {x - h} {x + h} \in N_x$.

Also, $\map {\omega_f} x$ is a lower bound for $\set {\map {\omega_f} {I' \cap D}: I' \in N_x}$ by the definition of $\map {\omega_f} x$.

Therefore:

$\map {\omega_f} {\openint {x - h} {x + h} \cap D} \ge \map {\omega_f} x$

We find:

\(\ds \map {\omega_f} x \le \map {\omega_f} {\openint {x - h} {x + h} \cap D}\)

\(\le\)

\(\ds \map {\omega_f} {I \cap D}\)

\(\ds \leadsto \ \ \)

\(\ds 0 \le \map {\omega_f} {\openint {x - h} {x + h} \cap D} - \map {\omega_f} x\)

\(\le\)

\(\ds \map {\omega_f} {I \cap D} - \map {\omega_f} x\)

\(\ds \leadsto \ \ \)

\(\ds 0 \le \map {\omega_f} {\openint {x - h} {x + h} \cap D} - \map {\omega_f} x\)

\(\le\)

\(\ds \map {\omega_f} {I \cap D} - \map {\omega_f} x < \epsilon\)

as $\map {\omega_f} {I \cap D} - \map {\omega_f} x < \epsilon$ is true

\(\ds \leadsto \ \ \)

\(\ds 0 \le \map {\omega_f} {\openint {x - h} {x + h} \cap D} - \map {\omega_f} x\)

\(<\)

\(\ds \epsilon\)

\(\ds \leadsto \ \ \)

\(\ds \size {\map {\omega_f} {\openint {x - h} {x + h} \cap D} - \map {\omega_f} x}\)

\(<\)

\(\ds \epsilon\)

which means that $\ds \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} \cap D}$ exists and equals $\map {\omega_f} x$ by the definition of limit.

In other words, $\map {\omega^L_f} x \in \R$ and $\map {\omega^L_f} x = \map {\omega_f} x$.

$\Box$

Sufficient Condition

Let $\map {\omega^L_f} x \in \R$.

We need to prove:

$\map {\omega_f} x \in \R$

$\map {\omega_f} x = \map {\omega^L_f} x$

where:

$\map {\omega_f} x = \inf \set {\map {\omega_f} {I \cap D}: I \in N_x}$

$\map {\omega_f} {I \cap D} = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$

$\map {\omega^L_f} x = \ds \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} \cap D}$

We have:

$\ds \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} \cap D} \in \R$ as $\map {\omega^L_f} x \in \R$

Therefore, $\map {\omega_f} {\openint {x - h} {x + h} \cap D} \in \R$ for a small enough $h$ in $\R_{>0}$ by the definition of limit.

Let $h$ be such a real number.

We observe that $\openint {x - h} {x + h}$ is a neighborhood in $N_x$.

We have:

$\openint {x - h} {x + h} \in N_x$

$\map {\omega_f} {\openint {x - h} {x + h} \cap D} \in \R$

Accordingly:

$\map {\omega_f} x \in \R$ by Infimum of Set of Oscillations on Set

$\map {\omega_f} x = \map {\omega^L_f} x$ follows by Lemma.

This finishes the proof of the theorem.

$\blacksquare$