The oscillation $\omega_f(S) = \sup\{\,|f(x) - f(y)| : x, y \in S\,\}$ of $f$ on a set and the oscillation $\omega_f(c) = \inf_{\delta > 0} \omega_f(A \cap N_\delta(c))$ at a point, both taken in the extended reals

Let $A \subseteq \mathbb{R}$ and let $f : A \to \mathbb{R}$. All suprema and infima below are taken in the extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$ (The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined), where every subset has a least upper bound and a greatest lower bound (Every subset of $\overline{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\overline{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$); no boundedness hypothesis on $f$ is therefore needed anywhere, and none is imposed.

Oscillation on a set. For $S \subseteq A$ put

$\omega_f(S) \;:=\; \sup\{\, |f(x) - f(y)| \;:\; x, y \in S \,\} \;\in\; \overline{\mathbb{R}} .$

Oscillation at a point. For $c \in A$ put

$\omega_f(c) \;:=\; \inf\{\, \omega_f(A \cap N_\delta(c)) \;:\; \delta \in \mathbb{R},\ \delta > 0 \,\} \;\in\; \overline{\mathbb{R}},$

where $N_\delta(c) = (c - \delta, c + \delta)$ is the $\delta$-neighbourhood of $c$ (The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-neighbourhood of a point of $\mathbb{R}$).

The two uses of the symbol $\omega_f$ are distinguished by their argument: a subset of $A$ in the first, a point of $A$ in the second. Where confusion is possible the first is written $\omega_f(S)$ with $S$ named as a set.

Both values are well posed; point oscillation and nonempty-set oscillation are nonnegative

The set in the first display is nonempty whenever $S$ is, since $x = y \in S$ gives the value $|f(x) - f(x)| = 0$; so $\omega_f(S) \ge 0$ for nonempty $S$, and $\omega_f(S) = \sup \varnothing = -\infty$ for $S = \varnothing$ (Every subset of $\overline{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\overline{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$). Only nonempty $S$ occurs below.

The set in the second display is nonempty, since some real $\delta > 0$ exists, and each of its members is $\ge 0$: for $c \in A$ the set $A \cap N_\delta(c)$ contains $c$ itself, because $|c - c| = 0 < \delta$, so it is nonempty and $\omega_f(A \cap N_\delta(c)) \ge 0$ (Basic properties of the absolute value). Hence $0$ is a lower bound of that set and

$0 \;\le\; \omega_f(c) \;\le\; \omega_f(A \cap N_\delta(c)) \qquad \text{for every real } \delta > 0,$

the second inequality because $\omega_f(c)$ is a lower bound of the set of which $\omega_f(A \cap N_\delta(c))$ is a member. In particular $\omega_f(c)$ is never $-\infty$.

Monotonicity, and the case of a bounded $f$

$\omega_f$ is monotone under inclusion. If $S \subseteq T \subseteq A$ then every value $|f(x) - f(y)|$ with $x, y \in S$ is also a value with $x, y \in T$, so the first set of values is contained in the second and $\omega_f(S) \le \omega_f(T)$: a supremum of a subset is at most the supremum of the set. Consequently $\delta \mapsto \omega_f(A \cap N_\delta(c))$ is nondecreasing in $\delta$, since $\delta \le \delta'$ gives $N_\delta(c) \subseteq N_{\delta'}(c)$ (The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-neighbourhood of a point of $\mathbb{R}$).

When $f$ is bounded, nonempty-set and point oscillations are real. Suppose there is a real $M$ with $|f(x)| \le M$ for every $x \in A$ (Lower bound, bounded below, bounded set). Then for $x, y \in A$,

$|f(x) - f(y)| \;\le\; |f(x)| + |f(y)| \;\le\; 2M$

(The triangle inequality, Basic properties of the absolute value), so $\omega_f(S) \le 2M$ for every $S \subseteq A$. If $S$ is nonempty, $\omega_f(S)$ is a real number in $[0,2M]$, and every point oscillation is also a real number in $[0,2M]$: the supremum of a nonempty subset of $\mathbb{R}$ that is bounded above in $\mathbb{R}$ is the real supremum (Every subset of $\overline{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\overline{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$, Complete ordered field (least-upper-bound property), Greatest lower bound (infimum)). The convention $\omega_f(\varnothing)=-\infty$ remains the single empty-set exception. Apart from that exception, an infinite extended value can occur only when $f$ is unbounded.

The notation. The letter is $\omega$ throughout this library, never "$\operatorname{osc}$", and the function is always in the subscript.