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 $\map {\omega_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\map {\omega_f} x = \displaystyle \inf \set {\map {\omega_f} I: I \in N_x}$ where $\map {\omega_f} I$ is the oscillation of $f$ on a real set $I$: :$\map {\omega_f} I = \displaystyle \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$ Let $\map {\omega^L_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\map {\omega^L_f} x = \displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$ Let $\map {\omega^L_f} x \in \R$. Let $\map {\omega_f} x \in \R$. Then $\map {\omega^L_f} x = \map {\omega_f} x$.  == Proof == We know that $\map {\omega^L_f} x$ and $\map {\omega_f} x$ are real numbers. We need to prove that $\map {\omega^L_f} x = \map {\omega_f} x$. Let $\epsilon \in \R_{>0}$. First, we aim to prove that $\size {\map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x} < \epsilon$ for a small enough $h \in R_{>0}$. $\map {\omega^L_f} x = \displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$ means by the definition of limit from the right that a strictly positive real number $h_1$ exists such that: :$\size {\map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega^L_f} x)} < \epsilon$ for every $h$ that satisfies: $0 < h < h_1$. This means in particular that $\map {\omega_f} {\openint {x - h} {x + h} } \in \R$ for every $h$ that satisfies: $0 < h < h_1$. Let $h'$ be a real number that satisfies: $0 < h' < h_1$. We observe that $\openint {x - h'} {x + h'} \in N_x$. Therefore, $\map {\omega_f} {\openint {x - h'} {x + h'} } \in \set {\map {\omega_f} I: I \in N_x}$. By definition, $\map {\omega_f} x$ is a lower bound for $\set {\map {\omega_f} I: I \in N_x}$. Accordingly: :$\map {\omega_f} {\openint {x - h'} {x + h'} } \ge \map {\omega_f} x$ The fact that $\map {\omega_f} x \in \R$ implies that: :$\map {\omega_f} I - \map {\omega_f} x < \epsilon$ 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 $\map {\omega_f} I \in \R$. 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 $\openint {x - h_2} {x + h_2}$ is a subset of $I$. Let $h''$ be a real number that satisfies: $0 < h'' < h_2$. We observe that $\openint {x - h''} {x + h''}$ is a subset of $I$. We have: :$\map {\omega_f} I \in \R$ :$\openint {x - h''} {x + h''}$ is a subset of $I$ Therefore: :$\map {\omega_f} {\openint {x - h''} {x + h''} } \le \map {\omega_f} I$ by Oscillation on Subset Putting all this together, we get for every $h$ that satisfies: $0 < h < \min \set {h_1, h_2}$: {{begin-eqn}} {{eqn | l = \map {\omega_f} {\openint {x - h} {x + h} }       | o = \le       | r = \map {\omega_f} I }} {{eqn | ll= \leadsto       | l = \map {\omega_f} x \le \map {\omega_f} {\openint {x - h} {x + h} }       | o = \le       | r = \map {\omega_f} I       | c = as $\map {\omega_f} x \le \map {\omega_f} {\openint {x - h} {x + h} }$ is true }} {{eqn | ll= \leadsto       | l = \map {\omega_f} x \le \map {\omega_f} {\openint {x - h} {x + h} }       | o = \le       | r = \map {\omega_f} I < \map {\omega_f} x + \epsilon       | c = as $\map {\omega_f} I < \map {\omega_f} x + \epsilon$ is true }} {{eqn | ll= \leadsto       | l = \map {\omega_f} x \le \map {\omega_f} {\openint {x - h} {x + h} }       | o = <       | r = \map {\omega_f} x + \epsilon       | c =  }} {{eqn | ll= \leadsto       | l = 0 \le \map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x       | o = <       | r = \epsilon }} {{eqn | ll= \leadsto       | l = \size {\map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x}       | o = <       | r = \epsilon }} {{end-eqn}} Thus, we achieved our first aim. Next, we get for every $h$ that satisfies: $0 < h < \min \set {h_1, h_2}$: {{begin-eqn}} {{eqn | l = \size {\map {\omega^L_f} x - \map {\omega_f} x}       | r = \size {\map {\omega^L_f} x - \map {\omega_f} {\openint {x - h} {x + h} } + \map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x} }} {{eqn | o = \le       | r = \size {\map {\omega^L_f} x - \map {\omega_f} {\openint {x - h} {x + h} } } + \size {\map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x}       | c = Triangle Inequality for Real Numbers }} {{eqn | o = <       | r = \epsilon + \epsilon }} {{eqn | r = 2 \epsilon }} {{end-eqn}} This holds for every $\epsilon \in \R_{>0}$. Therefore, $\size {\map {\omega^L_f} x - \map {\omega_f} x} = 0$ as $\size {\map {\omega^L_f} x - \map {\omega_f} x}$ is independent of $\epsilon$. Accordingly: :$\map {\omega^L_f} x = \map {\omega_f} x$  {{Improve|Implement a better category than this}}