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 open subset neighborhoods of $x$.

Let $\omega_f \left({x}\right)$ be the oscillation of $f$ at $x$ as defined by:


 * $\omega_f \left({x}\right) = \displaystyle \inf \left\{{\omega_f \left({I}\right): I \in N_x}\right\}$

where $\omega_f \left({I}\right)$ is the oscillation of $f$ on a real set $I$:


 * $\omega_f \left({I}\right) = \displaystyle \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$

Let $\omega^L_f \left({x}\right)$ be the oscillation of $f$ at $x$ as defined by:
 * $\omega^L_f \left({x}\right) = \displaystyle \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right)$

Then:


 * $\omega_f \left({x}\right) \in \R$ $\omega^L_f \left({x}\right) \in \R$

and, if $\omega_f \left({x}\right)$ and $\omega^L_f \left({x}\right)$ exist as real numbers:


 * $\omega_f \left({x}\right) = \omega^L_f \left({x}\right)$

Necessary Condition
Let $\omega_f \left({x}\right) \in \R$.

We need to prove:


 * $\omega^L_f \left({x}\right) \in \R$
 * $\omega^L_f \left({x}\right) = \omega_f \left({x}\right)$

where:
 * $\omega^L_f \left({x}\right) = \displaystyle \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right)$


 * $\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) = \displaystyle \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in \left({x - h \,.\,.\, x + h}\right) \cap D}\right\}$


 * $\omega_f \left({x}\right) = \displaystyle \inf \left\{{\omega_f \left({I}\right): I \in N_x}\right\}$


 * $\omega_f \left({I}\right) = \displaystyle \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$

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

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


 * $\omega_f \left({I}\right) - \omega_f \left({x}\right) < \epsilon$ 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 $\omega_f \left({I}\right) \in \R$.

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 $\left({x - \delta \,.\,.\, x + \delta}\right)$ is a subset of $I$.

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

We observe that $\left({x - h \,.\,.\, x + h}\right) \subset I$.

Also, $\left({x - h \,.\,.\, x + h}\right) \in N_x$.

We have:


 * $I \in N_x$


 * $\left({x - h \,.\,.\, x + h}\right) \in N_x$


 * $\left({x - h \,.\,.\, x + h}\right) \subset I$


 * $\omega_f \left({I}\right) \in \R$

from which follows by Oscillation on Subset:


 * $\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) \in \R$


 * $\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) \le \omega_f \left({I}\right)$

We have that $\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) \in \left\{{\omega_f \left({I'}\right): I' \in N_x}\right\}$ as $\left({x - h \,.\,.\, x + h}\right) \in N_x$.

Also, $\omega_f \left({x}\right)$ is a lower bound for $\left\{{\omega_f \left({I'}\right): I' \in N_x}\right\}$ by the definition of $\omega_f \left({x}\right)$.

Therefore:
 * $\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) \ge \omega_f \left({x}\right)$

We find:

which means that $\displaystyle \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right)$ exists and equals $\omega_f \left({x}\right)$ by the definition of limit.

In other words, $\omega^L_f \left({x}\right) \in \R$ and $\omega^L_f \left({x}\right) = \omega_f \left({x}\right)$.

Sufficient Condition
Let $\omega^L_f \left({x}\right) \in \R$.

We need to prove:


 * $\omega_f \left({x}\right) \in \R$


 * $\omega_f \left({x}\right) = \omega^L_f \left({x}\right)$

where:


 * $\omega_f \left({x}\right) = \displaystyle \inf \left\{{\omega_f \left({I}\right): I \in N_x}\right\}$


 * $\omega_f \left({I}\right) = \displaystyle \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$


 * $\omega^L_f \left({x}\right) = \displaystyle \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right)$

We have: $\displaystyle \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) \in \R$ as $\omega^L_f \left({x}\right) \in \R$.

Therefore, $\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) \in \R$ for a small enough $h$ in $\R_{>0}$ by the definition of limit.

Let $h$ be such a real number.

We observe that $\left({x - h \,.\,.\, x + h}\right)$ is a neighborhood in $N_x$.

We have:


 * $\left({x - h \,.\,.\, x + h}\right) \in N_x$


 * $\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) \in \R$

Accordingly:


 * $\omega_f \left({x}\right) \in \R$ by Infimum of Set of Oscillations on Set

$\omega_f \left({x}\right) = \omega^L_f \left({x}\right)$ follows by Lemma.

This finishes the proof of the theorem.