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 $x \in X$.

Then the following definitions of Oscillation at a Point are equivalent:

Definition 2
In these two definitions, 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
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 $\omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$ into the definition of $\omega_f \left({x}\right)$

Definition 2':


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

The theorem text of Oscillation at a Point equals Limit of Oscillation on a Set 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 neighborhoods of $x$ with the usual topology for the reals.


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

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:


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

We reformulate Definition a into Definition a' by:


 * changing set conditions into sup tags


 * renaming parameter set $I$ to $U$


 * substituting the definition of $\omega_f \left({I}\right)$ into the definition of $\omega_f \left({x}\right)$

Definition a':


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

An inspection of the definition of $\mathcal Nx'$ reveals that it uses the following concept of neighborhood of $x$:

A neighborhood of $x$ is a subset of $\R$ which contains (as a subset) an open set which itself contains (as an element) $x$.

This is the same definition of neighborhood that is used in the definition of $\mathcal N_x$ in Definition 1.

Therefore $\mathcal N_x' = \mathcal N_x$.

This allows us to reformulate Definition a' into Definition a'$\!$'.

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':


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

We observe:


 * 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

The definitions a and b are derived from the theorem text of Oscillation at a Point equals Limit of Oscillation on a Set.

Also, definitions a and b are equivalent by Oscillation at a Point equals Limit of Oscillation on a Set.

Therefore, definitions 1 and 2 are equivalent as definitions 1 and a are equivalent, definitions a and b are equivalent, and definitions b and 2 are equivalent.