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.
Continuity and derivative on a real interval
Definition
For , continuity at means that for every there is such that whenever and . Differentiability at an interior point means that the two-sided limit
exists in the real field; differentiability on means this holds at every interior point. For higher derivatives, define recursively on the points where the derivative exists. The class on an open interval means that these derivatives through order exist and are continuous there.
Depends on
Used by
- Finite-budget interval isolation for smooth scalar roots Algorithm
- Finite-budget scalar interval Newton root certification Algorithm
- Centered Taylor model with derivative-interval remainder Definition
- Smooth functions on the unit interval with bounded derivative data Definition
- A validated mass-action chemical equilibrium root Example
- An event root with derivative bounded away from zero Example
- Continuous functions have an oriented Riemann integral Lemma
- Continuous functions on a closed interval attain their extrema Lemma
- Explicit periodized Gaussian spread-tail bound Lemma
- Local uniform convergence of the periodized Gaussian Lemma
- Mean value theorem on a closed real interval Lemma
- PCHIP slopes are second-order accurate on uniform nonstationary grids Lemma
- Power series may be differentiated term by term inside their radius Lemma
- Rolle's theorem from attained extrema Lemma
- Sum, product, quotient, and chain rules from difference quotients Lemma
- Taylor's Lagrange remainder follows from repeated Rolle steps Lemma
- Taylor's theorem with an integral remainder Lemma
- The fundamental theorem of calculus and integration by parts Lemma
- PCHIP has third-order value accuracy on uniform nonstationary grids Theorem
- Scalar interval Newton exclusion and strict inclusion certificates Theorem
Dependency tree · one level
1 result within one dependency step 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
- Michael C. Colding, MIT 18.100B Real Analysis, Lecture 15: Derivatives; Laws for Differentiation (standard reference, not scraped)
- Michael C. Colding, MIT 18.100B Real Analysis, Lecture 16 (standard reference, not scraped)