How 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.
On a convex open set, a uniform bound implies
Statement
Let be convex and open, and let be totally differentiable at every point. If satisfies for every and , then
Facts & Assumptions
Given: The stated convex open domain, total differentiability, and uniform derivative bound.
A convex subset contains every line segment between two of its points (A convex subset of contains every line segment between two of its points).
The chain rule for total derivatives is (The chain rule for total derivatives: ).
The vector mean-value inequality gives when the derivative norm is bounded by (The mean value inequality: if is continuous and differentiable on with , then ).
Total differentiability implies continuity at the point of total differentiability (Total differentiability gives a local increment bound and therefore continuity).
Proof
If the conclusion is immediate. Otherwise put for ; [L1] keeps in .
The chain rule gives for , whose norm is at most by hypothesis.
By [L4] the curve is continuous at the endpoints, so [L3] applied on yields .
Depends on
- A convex subset of $\mathbb{R}^m$ contains every line segment between two of its points
- The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$
- The mean value inequality: if $f : [a,b] \to \mathbb{R}^m$ is continuous and differentiable on $(a,b)$ with $\lVert f'\rVert_2 \le M$, then $\lVert f(b)-f(a)\rVert_2 \le M(b-a)$
- The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\|h\|_2)$ remainder
- Total differentiability gives a local $O(\|h\|_2)$ increment bound and therefore continuity
Used by
- A totally differentiable map with zero derivative on a convex open set is constant Corollary
- A locally constant step map on the disconnected open set ℝ∖{0} has zero total derivative but is not globally Lipschitz Counterexample
- A C¹ map uniformly close to the identity derivative sandwiches a cube between contracted and expanded cubes Lemma
- Newton maps are uniform contractions near a point with invertible derivative Lemma
- On a small cube, a C¹ diffeomorphism distorts Jordan content by factors arbitrarily close to its linearized absolute determinant Lemma
- An injective C¹ map with invertible derivative sends compact Jordan sets to compact Jordan sets Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 115 results over 22 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis I, §8.4 (standard reference, not scraped)