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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
An inflection point as a point of continuity where convexity changes to concavity or conversely
Definition
Let and let be an interior point of the interval . The point is an inflection point when is continuous at (Continuity of at a point of and on : the - condition, its agreement with at a limit point, and continuity at an isolated point) and there is some such that, in either order, is convex but not concave on and concave but not convex on (Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval).
Remarks
The condition , even when defined, is neither part of this definition nor sufficient by itself: a change of shape is required.
Depends on
- Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval
- Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point
Used by
Dependency tree · two levels
9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- OpenStax Calculus Volume 1, §4.5 (standard reference, not scraped)