The oscillation ${\omega }_f(S)=\sup \{\mathrm{\mid }f(x)-f(y)\mathrm{\mid }:x,y\in S\}$ 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 $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, 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 $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, 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

${\omega }_f(S):=\sup \{\mathrm{\mid }f(x)-f(y)\mathrm{\mid }:x,y\in S\}\in \bar{\mathbb{R}}\mathrm{.}$

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

${\omega }_f(c):=\inf \{{\omega }_f(A\cap N_{\delta }(c)):\delta \in \mathbb{R},\delta >0\}\in \bar{\mathbb{R}},$

where $N_{\delta }(c)=(c-\delta ,c+\delta )$ is the $\delta$-neighbourhood of $c$ (The $\epsilon$-neighbourhood and the punctured $\epsilon$-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 $\mathrm{\mid }f(x)-f(x)\mathrm{\mid }=0$; so ${\omega }_f(S)\ge 0$ for nonempty $S$, and ${\omega }_f(S)=\sup \mathrm{\varnothing }=-\mathrm{\infty }$ for $S=\mathrm{\varnothing }$ (Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, 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 $\mathrm{\mid }c-c\mathrm{\mid }=0<\delta$, 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

$0\le {\omega }_f(c)\le {\omega }_f(A\cap N_{\delta }(c))\text{for every real }\delta >0,$

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 $-\mathrm{\infty }$.

Monotonicity, and the case of a bounded $f$

${\omega }_f$ is monotone under inclusion. If $S\subseteq T\subseteq A$ then every value $\mathrm{\mid }f(x)-f(y)\mathrm{\mid }$ 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 $\delta \le {\delta }^{\mathrm{\prime }}$ gives $N_{\delta }(c)\subseteq N_{{\delta }^{\mathrm{\prime }}}(c)$ (The $\epsilon$-neighbourhood and the punctured $\epsilon$-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 $\mathrm{\mid }f(x)\mathrm{\mid }\le M$ for every $x\in A$ (Lower bound, bounded below, bounded set). Then for $x,y\in A$,

$\mathrm{\mid }f(x)-f(y)\mathrm{\mid }\le \mathrm{\mid }f(x)\mathrm{\mid }+\mathrm{\mid }f(y)\mathrm{\mid }\le 2M$

(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 $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, 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 ${\omega }_f(\mathrm{\varnothing })=-\mathrm{\infty }$ 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 "$\mathrm{osc}$", and the function is always in the subscript.