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  "claimBoundary": "This page indexes Mathlib's finite-dimensional real normed-space form of Rademacher's theorem. With E carrying a measurable structure equal to its Borel structure, μ an additive Haar measure on E, E and F finite-dimensional over ℝ, and f : E → F satisfying LipschitzWith C f, the declaration concludes ∀ᵐ x ∂μ, DifferentiableAt ℝ f x. The endpoint is Fréchet differentiability at μ-almost every point, not differentiability everywhere, continuity of the derivative, continuous differentiability, or stronger smoothness.",
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  "exactFormalStatement": "theorem LipschitzWith.ae_differentiableAt {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] {C : ℝ≥0} {μ : Measure E} [FiniteDimensional ℝ E] [FiniteDimensional ℝ F] [IsAddHaarMeasure μ] {f : E → F} (h : LipschitzWith C f) : ∀ᵐ x ∂μ, DifferentiableAt ℝ f x",
  "family": "Geometric measure theory and nonsmooth calculus",
  "id": "library.mathlib.rademacher-theorem.v001",
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    "completeFormalType": "theorem LipschitzWith.ae_differentiableAt {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] {C : ℝ≥0} {μ : Measure E} [FiniteDimensional ℝ E] [FiniteDimensional ℝ F] [IsAddHaarMeasure μ] {f : E → F} (h : LipschitzWith C f) : ∀ᵐ x ∂μ, DifferentiableAt ℝ f x",
    "concepts": [
      "Lipschitz maps",
      "almost-everywhere differentiability",
      "Fréchet derivative",
      "additive Haar measure",
      "line derivatives",
      "Morrey's argument",
      "dense directions"
    ],
    "plainLanguage": "A globally Lipschitz map between finite-dimensional real normed spaces is Fréchet-differentiable at almost every point with respect to an additive Haar measure on its Borel domain.",
    "poster": {
      "alt": "An ivory Rademacher poster states the finite-dimensional real normed-space, Borel, additive Haar, and Lipschitz assumptions above a measured directional fabric whose regular points carry local linear ellipsoids while sparse muted points remain exceptional.",
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      "caption": "A globally Lipschitz map between finite-dimensional real normed spaces is Fréchet-differentiable at μ-almost every point.",
      "description": "The poster foregrounds the exact μ-almost-everywhere conclusion and keeps the finite-dimensional real normed-space, Borel measurable, additive Haar, and global Lipschitz assumptions visible. A central coordinate fabric distinguishes many regular points with direction fans and local linear-map ellipsoids from sparse exceptional points. Four source-bound movements summarize coordinate reduction, almost-everywhere line differentiability, Morrey linearity, and the dense-direction upgrade to a Fréchet derivative; the footer explicitly excludes everywhere differentiability and stronger smoothness.",
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      "generatedAt": "2026-07-27T10:31:58-04:00",
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      "sha256": "sha256:632424dc65d24a163b8f15507774732cacb5f9b7f47cfcd86b35287215e13d80",
      "title": "Rademacher's theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "ANALYSIS · DIFFERENTIABILITY\nRADEMACHER'S THEOREM\nE AND F FINITE-DIMENSIONAL REAL NORMED SPACES\nE HAS ITS BOREL MEASURABLE STRUCTURE · μ IS ADDITIVE HAAR\nf : E → F · LIPSCHITZ WITH CONSTANT C\nFOR μ-ALMOST EVERY x, f IS FRÉCHET-DIFFERENTIABLE AT x\nLIPSCHITZ CONTROL YIELDS FIRST-ORDER STRUCTURE ALMOST EVERYWHERE\nHOW THE CHECKED ROUTE MOVES\n1 · Reduce the range to real coordinates\n2 · Obtain line derivatives almost everywhere\n3 · Make them linear using Morrey's integral argument\n4 · Use dense directions to obtain a Fréchet derivative\nEXACT SCOPE\nAlmost everywhere, not everywhere. No continuity of the derivative or stronger smoothness."
    },
    "visual": {
      "alt": "A dark engraved directional fabric contains many regular starburst points and sparse dark exceptions, then gathers dense direction rays into norm-ball-to-ellipsoid linear maps and several distinct local affine tangent tiles.",
      "byteSize": 3174222,
      "caption": "Almost-everywhere compatible line derivatives on dense directions assemble into pointwise Fréchet derivatives under the Lipschitz bound.",
      "description": "A dark finite-dimensional coordinate fabric carries many ivory regular points and sparse dark exceptional pinpricks. Gold direction crowns gather around representative regular points, and emerald norm-ball-to-ellipsoid transformations encode continuous linear maps. Several separate local neighborhoods flatten toward differently oriented affine tiles, preserving the pointwise nature of the derivative and the theorem's almost-everywhere boundary.",
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      "generatedAt": "2026-07-27T10:31:58-04:00",
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      "mediaType": "image/png",
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      "publicPath": "/assets/library-theorems/rademacher-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:f3df02e8ffa4317d0a12e05f0443a55b69e5caec404edd5544eaefdc5fef09a2",
      "title": "Dense directional data assembling into almost-everywhere Fréchet derivatives",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Rademacher's theorem is a foundational bridge between metric regularity and differential structure: a global Lipschitz bound forces a first-order linear approximation at almost every point. Mathlib's Morrey-style route is especially instructive because it passes from one-dimensional line derivatives through almost-everywhere linearity and dense directions to the full Fréchet derivative."
  },
  "nonClaims": [
    "The declaration gives Fréchet differentiability at μ-almost every point; it does not claim differentiability at every point.",
    "The declaration does not claim that the derivative depends continuously on the point, that f is continuously differentiable, or that f has higher regularity.",
    "The selected theorem is finite-dimensional over ℝ and uses an additive Haar measure μ on the Borel domain E; it is not an infinite-dimensional or measure-free statement.",
    "ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence."
  ],
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  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.rademacher-theorem.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "A globally Lipschitz map between finite-dimensional real normed spaces is Fréchet-differentiable almost everywhere for an additive Haar measure on its domain.",
  "targetId": "target.library.mathlib.rademacher-theorem.v001",
  "title": "Rademacher's Theorem",
  "upstreamOrigin": {
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