Oscillation of a real function on subsets of $\mathbb{R}^m$ and at a point

Let $f:A\to\mathbb R$, $A\subseteq\mathbb R^m$. For $S\subseteq A$, define $\omega_f(S):=\sup_{\overline{\mathbb R}}\{|f(x)-f(y)|:x,y\in S\},$ with value $0$ when $S=\varnothing$. For $c\in A$, define $\omega_f(c):=\inf_{r>0}\omega_f(A\cap B(c,r)).$ The extended supremum exists by The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined and 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 $\mathbb{R}$; for bounded $f$ all values are finite. Balls are Open ball, closed ball and sphere in a metric space for the Euclidean metric ($\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it, Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page).

If $S\subseteq T$, then $\omega_f(S)\le\omega_f(T)$, directly from the supremum definition; hence the ball oscillations decrease as the radius shrinks and the infimum is well posed (Greatest lower bound (infimum), Basic properties of the absolute value). At $m=1$ this agrees with 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 on every nonempty set; only the empty-set convention differs, being $0$ here and $-\infty$ there.