Equivalence of Definitions of Oscillation at Point for Real Functions

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Theorem

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

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

Let $x \in X$.


The following definitions of the concept of Oscillation at a Point are equivalent:

Definition 1

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


The oscillation of $f$ at $x$ is defined as:

$\displaystyle \omega_f \left({x}\right) := \inf_{U \mathop \in \mathcal N_x} \omega_f \left({U \cap X}\right)$

where $\omega_f \left({U \cap X}\right)$ denotes the oscillation of $f$ on $U \cap X$.

Definition 2

The oscillation of $f$ at $x$ is defined as:

$\displaystyle \omega_f \left({x}\right) := \inf \left\{{\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right): \epsilon \in \R_{>0}}\right\}$

where $\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right)$ denotes the oscillation of $f$ on $\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X$.

Definition 3

The oscillation of $f$ at $x$ is defined as:

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

where $\omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$ denotes the oscillation of $f$ on $\left({x - h \,.\,.\, x + h}\right) \cap X$.


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

$\displaystyle \omega_f \left({A}\right) := \sup_{y, z \mathop \in A} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert$

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


Proof

Definitions 1 and 2 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':

$\displaystyle \omega_f \left({x}\right) := \inf_{U \mathop \in \mathcal N_x} \left({\sup_{y, z \mathop \in U \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$


We reformulate Definition 2 into Definition 2' by:

substituting the definition of $\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right)$ into the definition of $\omega_f \left({x}\right)$
changing set condition into sup tag

Definition 2':

$\displaystyle \omega_f \left({x}\right) := \inf_{\epsilon \mathop \in \R_{>0}} \left({\sup_{y, z \mathop \in \left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$


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 $\mathcal N_x$ be the set of open subset neighborhoods of $x$.

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

where:

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

Definition b:

$\omega^E_f \left({x}\right) = \inf \left\{{\omega_f \left({I}\right): I \in E_x}\right\}$

where:

$E_x$ is the set of $\epsilon$-neighborhoods of $x$


We reformulate Definition a into Definition a' by:

substituting the definition of $\omega_f \left({I}\right)$ into the definition of $\omega_f \left({x}\right)$
changing set conditions into sup tags
renaming parameter set $I$ to $U$

Definition a':

$\displaystyle \omega_f \left({x}\right) = \inf_{U \mathop \in \mathcal N_x} \left({\sup_{y, z \mathop \in U \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$


We reformulate Definition b into Definition b' by:

substituting the definition of $\omega_f \left({I}\right)$ into the definition of $\omega^E_f \left({x}\right)$
changing set conditions into sup tags
using $\epsilon$ as parameter instead of $I$

Definition b':

$\displaystyle \omega^E_f \left({x}\right) = \inf_{\epsilon \mathop \in \R_{>0}} \left({\sup_{y, z \mathop \in \left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X} \left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert}\right)$


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

Therefore:

definitions 1 and a are equivalent

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

Therefore:

definitions 2 and b are equivalent


We have:

definitions 1 and a are equivalent
definitions a and b are equivalent by Oscillation at Point (Infimum) equals Oscillation at Point (Epsilon-Neighborhood)
definitions b and 2 are equivalent

Therefore:

definitions 1 and 2 are equivalent

$\Box$


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':

$\displaystyle \omega_f \left({x}\right) := \inf_{U \mathop \in \mathcal N_x} \left({\sup_{y, z \mathop \in U \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$


We reformulate Definition 3 into Definition 3' by:

substituting the definition of $\omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$ into the definition of $\omega_f \left({x}\right)$

Definition 3':

$\displaystyle \omega_f \left({x}\right) := \lim_{h \mathop \to 0^+} \left({\sup_{y, z \mathop \in \left({x - h \,.\,.\, x + h}\right) \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$


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 $\mathcal N_x$ be the set of open subset neighborhoods of $x$.

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

where:

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

Definition b:

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


We reformulate Definition a into Definition a' by:

substituting the definition of $\omega_f \left({I}\right)$ into the definition of $\omega_f \left({x}\right)$
changing set conditions into sup tags
renaming parameter set $I$ to $U$

Definition a':

$\displaystyle \omega_f \left({x}\right) = \inf_{U \mathop \in \mathcal N_x} \left({\sup_{y, z \mathop \in U \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$


We reformulate Definition b into Definition b' by:

substituting the definition of $\omega_f \left({\left({x - h \,.\,.\, x + h}\right)}\right)$ (by using the definition of $\omega_f \left({I}\right)$) into the definition of $\omega_f \left({x}\right)$
changing set condition into sup tag

Definition b':

$\displaystyle \omega_f \left({x}\right) = \lim_{h \mathop \to 0^+} \left({\sup_{y, z \mathop \in \left({x - h \,.\,.\, x + h}\right) \cap X} \left\lvert{f \left({y}\right) - f \left({z}\right)}\right\rvert}\right)$


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

Therefore:

definitions 1 and a are equivalent

Also, definitions 3' and b' are the same, so they are equivalent.

Therefore:

definitions 3 and b are equivalent


We have:

definitions 1 and a are equivalent
definitions a and b are equivalent by Oscillation at Point (Infimum) equals Oscillation at Point (Limit)
definitions b and 3 are equivalent

Therefore:

definitions 1 and 3 are equivalent

$\Box$


Definitions 2 and 3 are equivalent

From:

Definitions 2 and 1 are equivalent
Definitions 1 and 3 are equivalent

follows:

Definitions 2 and 3 are equivalent

$\Box$


Thus all definitions listed in the theorem text are equivalent.

$\blacksquare$