Oscillation on Set is an Extended Real Number

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Theorem

Let $f: D \to \R$ be a real function where $D \subseteq \R$.

Let $x$ be a point in $D$.

Let $I$ be a real set that contains (as an element) $x$.

Let $\omega_f \left({I}\right)$ be the oscillation of $f$ on $I$:

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


Then:

$\omega_f \left({I}\right) \in \overline \R_{\ge 0}$

and:

$\omega_f \left({I}\right) = \begin{cases} \text{a positive real number} & \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\} \text{is bounded above} \\ \infty & \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\} \text{is not bounded above} \end{cases}$


Proof

We observe that $\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert = 0$ for $y = z = x$.

Therefore:

$0 \in \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ as $x \in I \cap D$

Accordingly:

$\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is non-empty


We have that $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is a real set as $\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert \in \R$ for every $y, z \in D$.

Every real number is less than $\infty$.

Therefore:

$\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is bounded above in $\overline \R$


There are two cases: either $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is bounded above (in $\R$), or it is not.


First, assume that:

$\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is bounded above (in $\R$)


We have that $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is bounded above and non-empty.

Therefore:

$\sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ exists as a real number by the Continuum Property


We know that $\sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is greater than or equal to every element of $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$.

Also $0 \in \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$.

Therefore, $\sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\} \ge 0$.

We also have that $\sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ exists as a real number.

Therefore, $\sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is a positive real number.

In other words:

$\omega_f \left({I}\right)$ is a positive real number as $\omega_f \left({I}\right) = \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$


Next, assume that:

$\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is not bounded above (in $\R$)


We have that $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is bounded above in $\overline \R$.

Therefore, $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ has $\infty$ as an upper bound.

A number that is less than $\infty$ is a real number or equal to $-\infty$.

No real number is an upper bound for $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ as $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is not bounded above in $\R$.

$-\infty$ is not an upper bound for $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ because then every real number would be an upper bound for $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$.

Therefore, $\infty$ is the only upper bound for $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$.

Accordingly, $\infty$ is the least upper bound of $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$.

In other words:

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


In either case, whether $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is bounded above or not:

$\omega_f \left({I}\right) \in \overline \R_{\ge 0}$.

$\blacksquare$