Alphabeta MathHow statement and proof provenance work
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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 is locally Lipschitz on compact coordinate subsets Lemma
- A C¹ map uniformly close to the identity derivative sandwiches a cube between contracted and expanded cubes Lemma
- A controlled nested horn construction embeds a closed three-ball Lemma
- A horn replacement block has an injective commutator meridian 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
- Every smooth manifold embeds in some finite-dimensional Euclidean space Theorem
Dependency tree · two levels
30 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, §8.4 (standard reference, not scraped)