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

== 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:
:$\map {\omega_f} x \in \R$ $\map {\omega^L_f} x \in \R$
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 ===
 
=== 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:
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 $\map {\omega^L_f} x = \map {\omega_f} x$.
=== 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.
Category:Real Analysis