On a convex open set, a uniform bound $\mathrm{\parallel }Df(z)v{\mathrm{\parallel }}_2\le M\mathrm{\parallel }v{\mathrm{\parallel }}_2$ implies $\mathrm{\parallel }f(y)-f(x){\mathrm{\parallel }}_2\le M\mathrm{\parallel }y-x{\mathrm{\parallel }}_2$

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 fixed RATTLE residual budget allows an O(epsilon) drift envelope Example
• A two-node semilinear diffusion FAS cycle has an explicit local rate Example
• The certified FAS contraction radius shrinks at a pitchfork Example
• The Coxeter complex of A₃: a triangulation of the sphere and the residue of a proper parabolic Example
• 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 Cᵏ⁺ʳ residual vanishing through order r-1 has a Cᵏ scaled extension Lemma
• A controlled nested horn construction embeds a closed three-ball Lemma
• A horn replacement block has an injective commutator meridian Lemma
• A parameterized C5 center manifold exists at a simple Hopf spectral split Lemma
• A small residual gives a nearby root under a uniform inverse and Lipschitz bound Lemma
• An implicit sensitivity evaluated at an approximate state has a local residual bound Lemma
• Holonomic phase residuals control distance to the consistent phase manifold Lemma
• Independent Brownian motion straightens a Poisson clock Lemma
• Modified Patankar Euler has a uniform first-order defect and one-step bound on positive tubes Lemma
• MPRK22(1) has a uniform second-order defect and one-step bound on positive tubes 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
• Parameter-dependent ODE flows have the variational tangent on a common tube Lemma
• The mollified Biot-Savart kernel is smooth and globally Lipschitz Lemma
• An equidimensional C¹ map sends null sets to null sets Proposition
• A bounded residual and derivative error give a local root certificate Theorem
• A globally Lipschitz neural ODE has a global smooth flow Theorem
• A hyperbolic fixed point of a local map is stable exactly when all multipliers contract Theorem
• A periodic C1 velocity has a global characteristic flow on a finite horizon Theorem
• An injective C¹ map with invertible derivative sends compact Jordan sets to compact Jordan sets Theorem
• Certified pseudo-arclength traces are locally sound and conditionally terminate Theorem
• Euler global state and log-density error on a bounded C² tube Theorem
• Every smooth manifold embeds in some finite-dimensional Euclidean space Theorem
• Exact RATTLE has second order global accuracy on compact tubes Theorem
• Finite fixed-point iteration sensitivity error bounds and the limits of residual control Theorem
• Forward Euler has uniform local order one and a one-step Lipschitz bound Theorem
• Residual bounds for exact and approximate-state adjoint hypergradients Theorem
• Scaled RATTLE residuals bound the finite solve trajectory error Theorem
• Smooth dependence of ODE solutions on parameters 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)