Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Oscillation of a real function on subsets of and at a point
Definition
Let , . For , define with value when . For , define The extended supremum exists by The extended real line , its order, and the arithmetic that is left undefined and Every subset of has a least upper bound and a greatest lower bound in , agreeing with the real supremum and infimum on nonempty sets bounded in ; for bounded all values are finite. Balls are Open ball, closed ball and sphere in a metric space for the Euclidean metric ( as the set of functions , and , , are metrics on it, Each is a norm on , and the induced metrics are exactly , and of the published metric-spaces page).
If , then , 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 this agrees with The oscillation of on a set and the oscillation at a point, both taken in the extended reals on every nonempty set; only the empty-set convention differs, being here and there.
Depends on
- 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
- The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined
- 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 $\mathbb{R}$
- Open ball, closed ball and sphere in a metric space
- $\mathbb{R}^n$ 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
- Lower bound, bounded below, bounded set
- Greatest lower bound (infimum)
- Basic properties of the absolute value
Used by
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Sources
- J. Lebl, Basic Analysis, Riemann Integral in Several Variables (standard reference, not scraped)
- J. Lebl, Basic Analysis, The Riemann-Lebesgue Criterion (standard reference, not scraped)