Equivalence of Definitions of Oscillation of Real Function at Point

Theorem
Let $X$ and $Y$ be real sets.

Let $f: X \to Y$ be a real function.

Let $x \in X$.

Definition 3
In the definitions above, the oscillation of $f$ on a non-empty set $A \subseteq X$ is defined as:


 * $\ds \map {\omega_f} A := \sup_{y, z \mathop \in A} \size {\map f y - \map f z}$

where the supremum is taken in the extended real numbers $\overline \R$.

Definitions 1 and 2 are equivalent
We reformulate Definition 1 into Definition 1' by:
 * substituting the definition of $\map {\omega_f} {U \cap X}$ into the definition of $\map {\omega_f} x$

Definition 1':
 * $\ds \map {\omega_f} x := \inf_{U \mathop \in \NN_x} \paren {\sup_{y, z \mathop \in U \cap X} \size {\map f y - \map f z} }$

We reformulate Definition 2 into Definition 2' by:
 * substituting the definition of $\map {\omega_f} {\openint {x - \epsilon} {x + \epsilon} \cap X}$ into the definition of $\map {\omega_f} x$
 * changing set condition into sup tag

Definition 2':
 * $\ds \map {\omega_f} x := \inf_{\epsilon \mathop \in \R_{>0} } \paren {\sup_{y, z \mathop \in \openint {x - \epsilon} {x + \epsilon} \cap X} \size {\map f y - \map f z} }$

The theorem text of Oscillation at Point (Infimum) equals Oscillation at Point (Epsilon-Neighborhood) also contains two definitions of Oscillation at a Point for a real function $f$.

We call them definitions $a)$ and $b)$.

Definition $a)$:

Let $\NN_x$ be the set of neighborhoods of $x$.


 * $\map {\omega_f} x = \ds \inf \set {\map {\omega_f} I: I \in \NN_x}$

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

Definition $b)$:
 * $\map {\omega^E_f} x = \inf \set {\map {\omega_f} I: I \in E_x}$

where:
 * $E_x$ is the set of $\epsilon$-neighborhoods of $x$

We reformulate Definition $a)$ into Definition $a')$ by:
 * substituting the definition of $\map {\omega_f} I$ into the definition of $\map {\omega_f} x$
 * changing set conditions into sup tags
 * renaming parameter set $I$ to $U$

Definition $a')$:
 * $\ds \map {\omega_f} x = \inf_{U \mathop \in \NN_x} \paren {\sup_{y, z \mathop \in U \cap X} \size {\map f y - \map f z} }$

We reformulate Definition $b)$ into Definition $b')$ by:
 * substituting the definition of $\map {\omega_f} I$ into the definition of $\map {\omega^E_f} x$
 * changing set conditions into sup tags
 * using $\epsilon$ as parameter instead of $I$

Definition $b')$:
 * $\ds \map {\omega^E_f} x = \inf_{\epsilon \mathop \in \R_{>0}} \paren {\sup_{y, z \mathop \in \openint {x - \epsilon} {x + \epsilon} \cap X} \size {\map f y - \map f z} }$

We observe that definitions $(1')$ and $a')$ are the same, so they are equivalent.

Therefore:
 * Definition 1 and $a)$ are equivalent

Also, definitions $(2')$ and $b')$ are the same, so they are equivalent.

Therefore:
 * Definition 2 and $b)$ are equivalent

We have:
 * Definition 1 and Definition $a)$ are equivalent
 * Definition $a)$ and Definition $b)$ are equivalent by Oscillation at Point (Infimum) equals Oscillation at Point (Epsilon-Neighborhood)
 * Definition $b)$ and Definition 2 are equivalent

Therefore:
 * Definition 1 and Definition 2 are equivalent

Definitions 1 and 3 are equivalent
We reformulate Definition 1 into Definition $(1')$ by:
 * substituting the definition of $\omega_f \left({U \cap X}\right)$ into the definition of $\omega_f \left({x}\right)$

Definition $(1')$:
 * $\ds \map {\omega_f} x := \inf_{U \mathop \in \NN_x} \paren {\sup_{y, z \mathop \in U \cap X} \size {\map f y - \map f z} }$

We reformulate Definition 3 into Definition $(3')$ by:
 * substituting the definition of $\map {\omega_f} {\openint {x - h} {x + h} \cap X}$ into the definition of $\map {\omega_f} x$

Definition 3':
 * $\ds \map {\omega_f} x := \lim_{h \mathop \to 0^+} \paren {\sup_{y, z \mathop \in \openint {x - h} {x + h} \cap X} \size {\map f y - \map f z} }$

The theorem text of Oscillation at Point (Infimum) equals Oscillation at Point (Limit) also contains two definitions of Oscillation at a Point for a real function $f$.

We call them definitions a and b.

Definition a:

Let $\NN_x$ be the set of open subset neighborhoods of $x$.


 * $\map {\omega_f} x = \ds \inf \set {\map {\omega_f} I: I \in \NN_x}$

where:
 * $\map {\omega_f} I = \ds \sup \set {\vert {\map f y - \map f z}: y, z \in I \cap X}$

Definition $b)$:
 * $\map {\omega_f} x = \ds \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$

We reformulate Definition $a)$ into Definition $a')$ by:
 * substituting the definition of $\map {\omega_f} I$ into the definition of $\map {\omega_f} x$
 * changing set conditions into sup tags
 * renaming parameter set $I$ to $U$

Definition $a')$:
 * $\ds \map {\omega_f} x = \inf_{U \mathop \in \NN_x} \paren {\sup_{y, z \mathop \in U \cap X} \size {\map f y - \map f z} }$

We reformulate Definition $b)$ into Definition $b')$ by:
 * substituting the definition of $\map {\omega_f} {\openint {x - h} {x + h} }$ (by using the definition of $\map {\omega_f} I$) into the definition of $\map {\omega_f} x$
 * changing set condition into sup tag

Definition $b')$:
 * $\ds \map {\omega_f} x = \lim_{h \mathop \to 0^+} \paren {\sup_{y, z \mathop \in \openint {x - h} {x + h} \cap X} \size {\map f y - \map f z} }$

We observe that definitions $(1')$ and $a')$ are the same, so they are equivalent.

Therefore:
 * Definition 1 and Definition $a)$ are equivalent

Also, Definition $(3')$ and Definition $b')$ are the same, so they are equivalent.

Therefore:
 * Definition 3 and Definition $b)$ are equivalent

We have:
 * Definition 1 and Definition $a)$ are equivalent
 * Definition $a)$ and Definition $b)$ are equivalent by Oscillation at Point (Infimum) equals Oscillation at Point (Limit)
 * Definition $b)$ and Definition 3 are equivalent

Therefore:
 * Definition 1 and Definition 3 are equivalent

Thus all definitions listed in the theorem text are equivalent.