{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "rademacher-euclidean-lipschitz-almost-everywhere-differentiability-1919",
    "title": "Rademacher proves almost-everywhere differentiability of Lipschitz functions",
    "summary": "Rademacher proves that a finite-dimensional Lipschitz function is totally differentiable outside a Lebesgue-null set. His original stronger sufficient condition uses a finite integrable upper local slope, and the paper connects almost-everywhere linearization to Jacobians and a change-of-variables formula for one-to-one transformations under its stated hypotheses.",
    "boundary": "The displayed proof is written for two variables, with extension to any finite number stated explicitly; finite-dimensional vector outputs follow componentwise, not arbitrary infinite-dimensional Banach targets. The derivative is a total linear approximation almost everywhere, not everywhere and not merely the separate existence of partial derivatives. The upper-local-slope variant requires its original finiteness/integrability conditions. The associated integral transformation is one-to-one and satisfies the original regularity/measurability assumptions, not a general noninjective multiplicity formula.",
    "area": "Real analysis and geometric measure theory",
    "topic": "Mathematics",
    "year": 1919,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "rademacher-euclidean-lipschitz-almost-everywhere-differentiability-1919",
    "target": {
      "record_id": "urn:uuid:4ec4e460-7368-45f6-a44c-51c842beacea",
      "version": 1,
      "content_hash": "sha256:9a6542b01cd261005ef23a5f31bf0d66e48f970661d81ab972462b6e11fbe6dc"
    },
    "humans": [
      "Hans Rademacher"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.",
    "agents": [],
    "related": [],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "cur20261020-rademacher-differenzierbarkeit-1919",
        "title": "Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale (Mathematische Annalen 79, 340–359)",
        "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0079/LOG_0039.pdf",
        "date": "1919"
      }
    ],
    "points": "300000",
    "published_grade": {
      "id": "rademacher-euclidean-lipschitz-almost-everywhere-differentiability-1919",
      "entry_id": "rademacher-euclidean-lipschitz-almost-everywhere-differentiability-1919",
      "band": "S",
      "modifier": "-",
      "target": {
        "record_id": "urn:uuid:4ec4e460-7368-45f6-a44c-51c842beacea",
        "version": 1,
        "content_hash": "sha256:9a6542b01cd261005ef23a5f31bf0d66e48f970661d81ab972462b6e11fbe6dc"
      },
      "what_changed": "Rademacher proves that a finite-dimensional Lipschitz function is totally differentiable outside a Lebesgue-null set. His original stronger sufficient condition uses a finite integrable upper local slope, and the paper connects almost-everywhere linearization to Jacobians and a change-of-variables formula for one-to-one transformations under its stated hypotheses.",
      "scope": "Rademacher proves that a finite-dimensional Lipschitz function is totally differentiable outside a Lebesgue-null set. His original stronger sufficient condition uses a finite integrable upper local slope, and the paper connects almost-everywhere linearization to Jacobians and a change-of-variables formula for one-to-one transformations under its stated hypotheses. The displayed proof is written for two variables, with extension to any finite number stated explicitly; finite-dimensional vector outputs follow componentwise, not arbitrary infinite-dimensional Banach targets. The derivative is a total linear approximation almost everywhere, not everywhere and not merely the separate existence of partial derivatives. The upper-local-slope variant requires its original finiteness/integrability conditions. The associated integral transformation is one-to-one and satisfies the original regularity/measurability assumptions, not a general noninjective multiplicity formula.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:4ec4e460-7368-45f6-a44c-51c842beacea",
        "version": 1,
        "content_hash": "sha256:9a6542b01cd261005ef23a5f31bf0d66e48f970661d81ab972462b6e11fbe6dc"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-23T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "Hans Rademacher",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale (Mathematische Annalen 79, 340–359)",
            "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0079/LOG_0039.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale (Mathematische Annalen 79, 340–359)",
            "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0079/LOG_0039.pdf"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The displayed proof is written for two variables, with extension to any finite number stated explicitly; finite-dimensional vector outputs follow componentwise, not arbitrary infinite-dimensional Banach targets. The derivative is a total linear approximation almost everywhere, not everywhere and not merely the separate existence of partial derivatives. The upper-local-slope variant requires its original finiteness/integrability conditions. The associated integral transformation is one-to-one and satisfies the original regularity/measurability assumptions, not a general noninjective multiplicity formula.",
        "domain": "Real analysis and geometric measure theory"
      },
      "sources": [
        {
          "citation": "Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale (Mathematische Annalen 79, 340–359)",
          "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0079/LOG_0039.pdf"
        }
      ],
      "statement": "# Rademacher proves almost-everywhere differentiability of Lipschitz functions\n\n## A rough graph can still have a local direction\n\nA curve may have a sharp corner and still be controlled: its output cannot change arbitrarily fast when its input changes a little. A Lipschitz condition gives a single constant bounding that change. Rademacher's theorem shows that, in finite-dimensional Euclidean space, this modest regularity already gives a total derivative almost everywhere. A total derivative is one linear approximation that works for all sufficiently small directions together, rather than a collection of unrelated directional guesses.\n\n## A corner you can inspect\n\nConsider the invented classroom function f(x)=the absolute value of x. Its output changes by no more than the change in its input, so it is Lipschitz with constant one. At a positive point, its derivative is one. At a negative point, the derivative is minus one. At zero, the two sides disagree. There is no derivative there. This example shows why an almost-everywhere conclusion can be strong without becoming an everywhere conclusion.\n\n## Almost everywhere has a precise meaning\n\nThe exceptional points form a set of Lebesgue measure zero. In one dimension, that means they can be covered by intervals of arbitrarily small total length. It does not necessarily mean that there are only finitely many exceptions. In higher dimensions the relevant measure is the corresponding volume. The theorem guarantees that the linear description exists outside a null set, even though the function need not have a continuous derivative or a visually smooth formula.\n\n## The original scope remains visible\n\nThe paper writes its argument for two variables and explicitly states the extension to any finite number. Its stronger local-slope condition keeps its own finiteness and integrability assumptions. The associated change-of-variables discussion has a one-to-one transformation and additional conditions. It is not a blanket statement for arbitrary infinite-dimensional targets, noninjective transformations or every weakly regular map. Separate existence of partial derivatives alone is also a weaker property than total differentiability.\n\n## Use numerical checks as illustrations\n\nAsk your agent to estimate difference quotients for absolute value at a positive point, a negative point and zero, using several decreasing step sizes. Report both left and right quotients. Then explain why sampling those three locations neither proves the theorem nor locates every possible exception for another function. A helpful result connects the observed example to the exact hypothesis and distinguishes a local numerical illustration from a universal analytical guarantee.\n\n## Scientific scope\n\nThe displayed proof is written for two variables, with extension to any finite number stated explicitly; finite-dimensional vector outputs follow componentwise, not arbitrary infinite-dimensional Banach targets. The derivative is a total linear approximation almost everywhere, not everywhere and not merely the separate existence of partial derivatives. The upper-local-slope variant requires its original finiteness/integrability conditions. The associated integral transformation is one-to-one and satisfies the original regularity/measurability assumptions, not a general noninjective multiplicity formula.\n\n## Sources for this explanation\n\n- [Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale (Mathematische Annalen 79, 340–359)](https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0079/LOG_0039.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "Rademacher proves that a finite-dimensional Lipschitz function is totally differentiable outside a Lebesgue-null set. His original stronger sufficient condition uses a finite integrable upper local slope, and the paper connects almost-everywhere linearization to Jacobians and a change-of-variables formula for one-to-one transformations under its stated hypotheses.",
      "title": "Rademacher proves almost-everywhere differentiability of Lipschitz functions"
    },
    "record_id": "urn:uuid:4ec4e460-7368-45f6-a44c-51c842beacea",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:4ec4e460-7368-45f6-a44c-51c842beacea",
      "version": 1,
      "content_hash": "sha256:9a6542b01cd261005ef23a5f31bf0d66e48f970661d81ab972462b6e11fbe6dc"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:b9671736637c6902d2ef51386bdd7dbad70c1717675dfba1d8f7db8e4e674694"
  },
  "target": {
    "record_id": "urn:uuid:4ec4e460-7368-45f6-a44c-51c842beacea",
    "version": 1,
    "content_hash": "sha256:9a6542b01cd261005ef23a5f31bf0d66e48f970661d81ab972462b6e11fbe6dc"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "rademacher-euclidean-lipschitz-almost-everywhere-differentiability-1919",
    "entry_id": "rademacher-euclidean-lipschitz-almost-everywhere-differentiability-1919",
    "revision": 1,
    "title": "Rademacher proves almost-everywhere differentiability of Lipschitz functions",
    "context": "Real analysis and geometric measure theory",
    "track": "Exact theorem and reusable mathematical method",
    "band": "S",
    "modifier": "-",
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-23",
    "evidence_cutoff": "2026-10-23",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Rademacher proves that a finite-dimensional Lipschitz function is totally differentiable outside a Lebesgue-null set. His original stronger sufficient condition uses a finite integrable upper local slope, and the paper connects almost-everywhere linearization to Jacobians and a change-of-variables formula for one-to-one transformations under its stated hypotheses.",
    "scope": "Rademacher proves that a finite-dimensional Lipschitz function is totally differentiable outside a Lebesgue-null set. His original stronger sufficient condition uses a finite integrable upper local slope, and the paper connects almost-everywhere linearization to Jacobians and a change-of-variables formula for one-to-one transformations under its stated hypotheses. The displayed proof is written for two variables, with extension to any finite number stated explicitly; finite-dimensional vector outputs follow componentwise, not arbitrary infinite-dimensional Banach targets. The derivative is a total linear approximation almost everywhere, not everywhere and not merely the separate existence of partial derivatives. The upper-local-slope variant requires its original finiteness/integrability conditions. The associated integral transformation is one-to-one and satisfies the original regularity/measurability assumptions, not a general noninjective multiplicity formula.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The displayed proof is written for two variables, with extension to any finite number stated explicitly; finite-dimensional vector outputs follow componentwise, not arbitrary infinite-dimensional Banach targets. The derivative is a total linear approximation almost everywhere, not everywhere and not merely the separate existence of partial derivatives. The upper-local-slope variant requires its original finiteness/integrability conditions. The associated integral transformation is one-to-one and satisfies the original regularity/measurability assumptions, not a general noninjective multiplicity formula.",
    "references": [
      {
        "title": "Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale (Mathematische Annalen 79, 340–359)",
        "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0079/LOG_0039.pdf",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      }
    ],
    "roles": [
      {
        "who": "The named authors in the research record",
        "credit": "SCIENCE",
        "did": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred."
      }
    ],
    "target": {
      "record_id": "urn:uuid:4ec4e460-7368-45f6-a44c-51c842beacea",
      "version": 1,
      "content_hash": "sha256:9a6542b01cd261005ef23a5f31bf0d66e48f970661d81ab972462b6e11fbe6dc"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "300000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": {
    "id": "rademacher-euclidean-lipschitz-almost-everywhere-differentiability-1919",
    "target": {
      "record_id": "urn:uuid:4ec4e460-7368-45f6-a44c-51c842beacea",
      "version": 1,
      "content_hash": "sha256:9a6542b01cd261005ef23a5f31bf0d66e48f970661d81ab972462b6e11fbe6dc"
    },
    "sections": [
      {
        "title": "A rough graph can still have a local direction",
        "text": "A curve may have a sharp corner and still be controlled: its output cannot change arbitrarily fast when its input changes a little. A Lipschitz condition gives a single constant bounding that change. Rademacher's theorem shows that, in finite-dimensional Euclidean space, this modest regularity already gives a total derivative almost everywhere. A total derivative is one linear approximation that works for all sufficiently small directions together, rather than a collection of unrelated directional guesses."
      },
      {
        "title": "A corner you can inspect",
        "text": "Consider the invented classroom function f(x)=the absolute value of x. Its output changes by no more than the change in its input, so it is Lipschitz with constant one. At a positive point, its derivative is one. At a negative point, the derivative is minus one. At zero, the two sides disagree. There is no derivative there. This example shows why an almost-everywhere conclusion can be strong without becoming an everywhere conclusion."
      },
      {
        "title": "Almost everywhere has a precise meaning",
        "text": "The exceptional points form a set of Lebesgue measure zero. In one dimension, that means they can be covered by intervals of arbitrarily small total length. It does not necessarily mean that there are only finitely many exceptions. In higher dimensions the relevant measure is the corresponding volume. The theorem guarantees that the linear description exists outside a null set, even though the function need not have a continuous derivative or a visually smooth formula."
      },
      {
        "title": "The original scope remains visible",
        "text": "The paper writes its argument for two variables and explicitly states the extension to any finite number. Its stronger local-slope condition keeps its own finiteness and integrability assumptions. The associated change-of-variables discussion has a one-to-one transformation and additional conditions. It is not a blanket statement for arbitrary infinite-dimensional targets, noninjective transformations or every weakly regular map. Separate existence of partial derivatives alone is also a weaker property than total differentiability."
      },
      {
        "title": "Use numerical checks as illustrations",
        "text": "Ask your agent to estimate difference quotients for absolute value at a positive point, a negative point and zero, using several decreasing step sizes. Report both left and right quotients. Then explain why sampling those three locations neither proves the theorem nor locates every possible exception for another function. A helpful result connects the observed example to the exact hypothesis and distinguishes a local numerical illustration from a universal analytical guarantee."
      }
    ],
    "references": [
      {
        "title": "Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale (Mathematische Annalen 79, 340–359)",
        "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0079/LOG_0039.pdf",
        "locator": "Pinned original contribution and stated hypotheses or experimental limits."
      }
    ]
  },
  "reading_status": "EXPLANATORY_GUIDE"
}