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.
Higher derivatives and the classes and
Definition
Let be an interval and . Put . Recursively, wherever is differentiable, put , with derivatives at endpoints understood in the one-sided sense fixed by The derivative of at a point that is a limit point of , and differentiability on a set and The left and right limits of at , as limits of the restrictions of to and .
For , the function is -times differentiable on if exists on for every . It is of class on if these derivatives exist and every , , is continuous on (Continuity of at a point of and on : the - condition, its agreement with at a limit point, and continuity at an isolated point). It is smooth, or , if it is for every .
Since (The natural numbers (von Neumann)), means continuity. The definitions also give . Existence of alone does not assert that is continuous.
Depends on
- The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set
- 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
- The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$
- The natural numbers $\mathbb{N}$ (von Neumann)
Used by
- A continuous function whose second derivative has opposite signs on the two sides of a point has an inflection point there Corollary
- A twice-differentiable function on an open interval is convex if and only if its second derivative is nonnegative Corollary
- A uniform limit of smooth functions need not be differentiable anywhere Corollary
- A flat smooth real function has no holomorphic extension near zero Counterexample
- A smooth function not equal to its Maclaurin series Counterexample
- Taylor and Maclaurin series Definition
- Taylor polynomials and their remainders Definition
- A bounded C¹ periodic oscillator made from a quartic Hermite spline Example
- For every k≥0, xᵏ|x| is Cᵏ but not Cᵏ⁺¹ Example
- The real function 1/(1+x²) is smooth on the real line but its Maclaurin series has radius one Example
- x↦ x³ changes from concave to convex at zero and has an inflection point there Example
- Higher-order Rolle theorem Lemma
- Scope, endpoint, factorial, and deferred-remainder conventions Remark
- Standard Maclaurin expansions Theorem
- Taylor-series representation by vanishing remainders Theorem
- The general Leibniz rule for the n-th derivative of a product Theorem
- The second-derivative test for strict local extrema Theorem
Dependency tree · two levels
18 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
- J. Lebl, Basic Analysis I, Taylor's theorem and related calculus (standard reference, not scraped)
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)