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

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

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

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

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

Proof
We know that $\omega^L_f \left({x}\right)$ and $\omega_f \left({x}\right)$ are real numbers.

We need to prove that $\omega^L_f \left({x}\right) = \omega_f \left({x}\right)$.

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

First, we aim to prove that $\left\vert{\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) - \omega_f \left({x}\right)}\right\vert < \epsilon$ for a small enough $h \in R_{>0}$.

$\omega^L_f \left({x}\right) = \displaystyle \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right)$ means by the definition of limit that a strictly positive real number $h_1$ exists such that:


 * $\left\vert{\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) - \omega^L_f \left({x}\right)}\right\vert < \epsilon$

for every $h$ that satisfies: $0 < h < h_1$.

This means in particular that $\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right) \in \R$ for every $h$ that satisfies: $0 < h < h_1$.

Let $h'$ be a real number that satisfies: $0 < h' < h_1$.

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

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

By definition, $\omega_f \left({x}\right)$ is a lower bound for $\left\{{\omega_f \left({I}\right): I \in N_x}\right\}$.

Accordingly:


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

The fact that $\omega_f \left({x}\right) \in \R$ implies that:


 * $\omega_f \left({I}\right) - \omega_f \left({x}\right) < \epsilon$ by Infimum of Set of Oscillations on Set is Arbitrarily Close

for an $I \in N_x$.

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 number $h_2 \in \R_{>0}$ exists such that $\left({x - h_2 \,.\,.\, x + h_2}\right)$ is a subset of $I$.

Let $h$ be a real number that satisfies: $0 < h < h_2$.

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

We have:


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


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

Therefore:


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

Putting all this together, we get for every $h$ that satisfies: $0 < h < \min \left({h_1, h_2}\right)$:

Thus, we achieved our first aim.

Next, we get for every $h$ that satisfies: $0 < h < \min \left({h_1, h_2}\right)$:

This holds for every $\epsilon \in \R_{>0}$.

Therefore, $\left\vert{\omega^L_f \left({x}\right) - \omega_f \left({x}\right)}\right\vert = 0$ as $\left\vert{\omega^L_f \left({x}\right) - \omega_f \left({x}\right)}\right\vert$ is independent of $\epsilon$.

Accordingly, $\omega^L_f \left({x}\right) = \omega_f \left({x}\right)$.