Oscillation on Subset

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

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

Let $S_x$ be a set of real sets that contain (as an element) $x$.

Let $\omega_f \left({I}\right)$ be the oscillation of $f$ on a set $I$ in $S_x$:
 * $\omega_f \left({I}\right) = \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$

Let $I \in S_x$.

Let $\omega_f \left({I}\right) \in \R$.

Let $J \in S_x$ be a subset of $I$.

Then:
 * $\omega_f \left({J}\right) \in \R$
 * $\omega_f \left({J}\right) \le \omega_f \left({I}\right)$

Proof
Let:
 * $I, J \in S_x$
 * $J \subset I$
 * $\omega_f \left({I}\right) \in \R$

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

We need to prove:
 * $\omega_f \left({J}\right) \in \R$
 * $\omega_f \left({J}\right) \le \omega_f \left({I}\right)$

We intend to prove that $\omega_f \left({J}\right) \in \R$.

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

We have that $J$ is a subset of $I$.

Therefore:
 * $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is a subset of $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$

The statement $\omega_f \left({I}\right) \in \R$ means that $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ admits a supremum.

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

Accordingly:
 * $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is bounded above as $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is a subset of $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$

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

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

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

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

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

Therefore, $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ admits a supremum by the Continuum Property.

In other words:

We finished proving that $\omega_f \left({J}\right) \in \R$.

It remains to prove that $\omega_f \left({J}\right) \le \omega_f \left({I}\right)$.

We have:
 * $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is a subset of $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$
 * $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ admits a supremum
 * $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ admits a supremum

Then: