{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "darboux-intermediate-value-property-derivatives-1875",
    "title": "Darboux: a derivative can be discontinuous and still skip no intermediate slope",
    "summary": "A derivative measures slope. It is tempting to assume that slopes must change continuously, but this is false. Darboux proves a different constraint: if a finite derivative exists at every point of an interval, it takes every value between any two of its values. The proof subtracts a line with an intermediate slope. The resulting function has an interior extremum, where its derivative vanishes, so the original derivative takes the desired slope. The same paper gives examples showing why this intermediate-value property can survive even when the derivative is discontinuous.",
    "boundary": "The derivative must exist finitely throughout the interval, not merely almost everywhere. The intermediate-value property does not imply continuity, and a function with that property need not be a derivative. The subsequent primitive-function criterion additionally assumes integrability. This record covers the derivative theorem, not ownership of all discontinuous-function examples, Riemann integration or the mean-value theorem. The manuscript is signed in1874; the journal volume was published in1875.",
    "area": "Real analysis",
    "topic": "Mathematics",
    "year": 1875,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "darboux-intermediate-value-property-derivatives-1875",
    "target": {
      "record_id": "urn:uuid:a572a20b-6642-435c-a2d7-747a12e7faa6",
      "version": 1,
      "content_hash": "sha256:27d669f8f51cd2afe17f69d5b1ff3649dbce8db90c77d8de8422867a62c94de5"
    },
    "humans": [
      "Gaston Darboux"
    ],
    "authors_complete": true,
    "human_role_note": "Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record 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": "darboux-derivative-intermediate-value-original1875",
        "title": "Mémoire sur les fonctions discontinues — original 1875 journal scan, especially §IX pp.109–112",
        "url": "https://www.numdam.org/item/10.24033/asens.122.pdf",
        "date": "1875"
      }
    ],
    "geography": {
      "country_codes": [
        "FR"
      ],
      "location_basis": "Original author affiliation is the Ecole Normale in the French memoir. Present-day place labels do not assert nationality,modern-state ownership or historical territorial jurisdiction.",
      "source_url": "https://www.numdam.org/item/10.24033/asens.122.pdf",
      "target": {
        "record_id": "urn:uuid:a572a20b-6642-435c-a2d7-747a12e7faa6",
        "version": 1,
        "content_hash": "sha256:27d669f8f51cd2afe17f69d5b1ff3649dbce8db90c77d8de8422867a62c94de5"
      },
      "basis": "RESEARCH_LOCATION_OR_PAPER_AFFILIATION",
      "reviewed_on": "2026-10-23",
      "nationality_inferred": false,
      "sources": [
        {
          "title": "Mémoire sur les fonctions discontinues — original 1875 journal scan, especially §IX pp.109–112",
          "url": "https://www.numdam.org/item/10.24033/asens.122.pdf"
        }
      ],
      "coverage_status": "COMPLETE",
      "country_labels": {
        "FR": "France"
      }
    },
    "points": "220000",
    "published_grade": {
      "id": "darboux-intermediate-value-property-derivatives-1875",
      "entry_id": "darboux-intermediate-value-property-derivatives-1875",
      "band": "A",
      "modifier": "+",
      "target": {
        "record_id": "urn:uuid:a572a20b-6642-435c-a2d7-747a12e7faa6",
        "version": 1,
        "content_hash": "sha256:27d669f8f51cd2afe17f69d5b1ff3649dbce8db90c77d8de8422867a62c94de5"
      },
      "what_changed": "A derivative measures slope. It is tempting to assume that slopes must change continuously, but this is false. Darboux proves a different constraint: if a finite derivative exists at every point of an interval, it takes every value between any two of its values. The proof subtracts a line with an intermediate slope. The resulting function has an interior extremum, where its derivative vanishes, so the original derivative takes the desired slope. The same paper gives examples showing why this intermediate-value property can survive even when the derivative is discontinuous.",
      "scope": "A derivative measures slope. It is tempting to assume that slopes must change continuously, but this is false. Darboux proves a different constraint: if a finite derivative exists at every point of an interval, it takes every value between any two of its values. The proof subtracts a line with an intermediate slope. The resulting function has an interior extremum, where its derivative vanishes, so the original derivative takes the desired slope. The same paper gives examples showing why this intermediate-value property can survive even when the derivative is discontinuous. The derivative must exist finitely throughout the interval, not merely almost everywhere. The intermediate-value property does not imply continuity, and a function with that property need not be a derivative. The subsequent primitive-function criterion additionally assumes integrability. This record covers the derivative theorem, not ownership of all discontinuous-function examples, Riemann integration or the mean-value theorem. The manuscript is signed in1874; the journal volume was published in1875.",
      "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:a572a20b-6642-435c-a2d7-747a12e7faa6",
        "version": 1,
        "content_hash": "sha256:27d669f8f51cd2afe17f69d5b1ff3649dbce8db90c77d8de8422867a62c94de5"
      },
      "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": "Gaston Darboux",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Mémoire sur les fonctions discontinues — original 1875 journal scan, especially §IX pp.109–112",
            "url": "https://www.numdam.org/item/10.24033/asens.122.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Mémoire sur les fonctions discontinues — original 1875 journal scan, especially §IX pp.109–112",
            "url": "https://www.numdam.org/item/10.24033/asens.122.pdf"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The derivative must exist finitely throughout the interval, not merely almost everywhere. The intermediate-value property does not imply continuity, and a function with that property need not be a derivative. The subsequent primitive-function criterion additionally assumes integrability. This record covers the derivative theorem, not ownership of all discontinuous-function examples, Riemann integration or the mean-value theorem. The manuscript is signed in1874; the journal volume was published in1875.",
        "domain": "Real analysis"
      },
      "sources": [
        {
          "citation": "Mémoire sur les fonctions discontinues — original 1875 journal scan, especially §IX pp.109–112",
          "url": "https://www.numdam.org/item/10.24033/asens.122.pdf"
        }
      ],
      "statement": "# Darboux: a derivative can be discontinuous and still skip no intermediate slope\n\n## Can a slope skip a value?\n\nFor a smooth-looking curve, it seems natural that slopes change continuously. But differentiability of a function does not force continuity of its derivative. Darboux identifies a different universal constraint: when the finite derivative exists everywhere in an interval, it cannot skip an intermediate value. If its slope is one at one location and three at another, the slope must be two somewhere between them. This remains true even when the derivative is discontinuous.\n\n## Subtract the slope you want to find\n\nChoose a desired intermediate slope m and subtract the straight line m times x from the original function. At the two selected endpoints, the new derivative has opposite signs. In the appropriate orientation the new function begins by increasing and ends by decreasing, so its maximum occurs at an interior point. At that maximum the derivative is zero. Undoing the subtraction shows that the original derivative equals m there. The argument turns a slope question into an extremum question.\n\n## Oscillation differs from a jump\n\nDefine a classroom function as x squared times sin of one divided by x away from zero, and set its value at zero to zero. Its derivative at zero is zero, while away from zero the derivative contains a cosine term that keeps oscillating. The derivative is discontinuous at zero, yet it does not create a clean jump that skips all intermediate values. The example shows why discontinuous is a broader category than having a simple step discontinuity.\n\n## Do not reverse the implication\n\nEvery derivative in the stated scope has the intermediate-value property. It does not follow that every function with that property is a derivative. Nor may the hypothesis be weakened silently to a derivative existing only almost everywhere. A step function illustrates the importance of the condition: it can jump from one level to another without taking intermediate levels, so it cannot be an everywhere-existing finite derivative on that interval. The theorem provides a necessary condition, not a complete characterization.\n\n## An attainable source-and-example check\n\nAsk your agent to differentiate the oscillating example away from zero and compute the derivative at zero from its definition. It should explain why a graph sampled at finitely many points cannot establish continuity or the intermediate-value property by itself. Then have it identify the exact derivative hypothesis in the source. The manuscript signature is from1874and the journal publication from1875; those dates serve different roles. A concise checked explanation is a useful contribution without claiming a new theorem.\n\n## Scientific scope\n\nThe derivative must exist finitely throughout the interval, not merely almost everywhere. The intermediate-value property does not imply continuity, and a function with that property need not be a derivative. The subsequent primitive-function criterion additionally assumes integrability. This record covers the derivative theorem, not ownership of all discontinuous-function examples, Riemann integration or the mean-value theorem. The manuscript is signed in1874; the journal volume was published in1875.\n\n## Sources for this explanation\n\n- [Mémoire sur les fonctions discontinues — original 1875 journal scan, especially §IX pp.109–112](https://www.numdam.org/item/10.24033/asens.122.pdf): Original SectionIXpp109–112,oscillating derivative example and subtract-a-line proof;title/history introductory pages.\n",
      "summary": "A derivative measures slope. It is tempting to assume that slopes must change continuously, but this is false. Darboux proves a different constraint: if a finite derivative exists at every point of an interval, it takes every value between any two of its values. The proof subtracts a line with an intermediate slope. The resulting function has an interior extremum, where its derivative vanishes, so the original derivative takes the desired slope. The same paper gives examples showing why this intermediate-value property can survive even when the derivative is discontinuous.",
      "title": "Darboux: a derivative can be discontinuous and still skip no intermediate slope"
    },
    "record_id": "urn:uuid:a572a20b-6642-435c-a2d7-747a12e7faa6",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:a572a20b-6642-435c-a2d7-747a12e7faa6",
      "version": 1,
      "content_hash": "sha256:27d669f8f51cd2afe17f69d5b1ff3649dbce8db90c77d8de8422867a62c94de5"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:8cb7c8bba6dd2766ca7fe9787723b9ef83a9c4618d5077f1435837500adc3d73"
  },
  "target": {
    "record_id": "urn:uuid:a572a20b-6642-435c-a2d7-747a12e7faa6",
    "version": 1,
    "content_hash": "sha256:27d669f8f51cd2afe17f69d5b1ff3649dbce8db90c77d8de8422867a62c94de5"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "darboux-intermediate-value-property-derivatives-1875",
    "entry_id": "darboux-intermediate-value-property-derivatives-1875",
    "revision": 1,
    "title": "Darboux: a derivative can be discontinuous and still skip no intermediate slope",
    "context": "Real analysis",
    "track": "Exact theorem and reusable mathematical method",
    "band": "A",
    "modifier": "+",
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-23",
    "evidence_cutoff": "2026-10-23",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "A derivative measures slope. It is tempting to assume that slopes must change continuously, but this is false. Darboux proves a different constraint: if a finite derivative exists at every point of an interval, it takes every value between any two of its values. The proof subtracts a line with an intermediate slope. The resulting function has an interior extremum, where its derivative vanishes, so the original derivative takes the desired slope. The same paper gives examples showing why this intermediate-value property can survive even when the derivative is discontinuous.",
    "scope": "A derivative measures slope. It is tempting to assume that slopes must change continuously, but this is false. Darboux proves a different constraint: if a finite derivative exists at every point of an interval, it takes every value between any two of its values. The proof subtracts a line with an intermediate slope. The resulting function has an interior extremum, where its derivative vanishes, so the original derivative takes the desired slope. The same paper gives examples showing why this intermediate-value property can survive even when the derivative is discontinuous. The derivative must exist finitely throughout the interval, not merely almost everywhere. The intermediate-value property does not imply continuity, and a function with that property need not be a derivative. The subsequent primitive-function criterion additionally assumes integrability. This record covers the derivative theorem, not ownership of all discontinuous-function examples, Riemann integration or the mean-value theorem. The manuscript is signed in1874; the journal volume was published in1875.",
    "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 derivative must exist finitely throughout the interval, not merely almost everywhere. The intermediate-value property does not imply continuity, and a function with that property need not be a derivative. The subsequent primitive-function criterion additionally assumes integrability. This record covers the derivative theorem, not ownership of all discontinuous-function examples, Riemann integration or the mean-value theorem. The manuscript is signed in1874; the journal volume was published in1875.",
    "references": [
      {
        "title": "Mémoire sur les fonctions discontinues — original 1875 journal scan, especially §IX pp.109–112",
        "url": "https://www.numdam.org/item/10.24033/asens.122.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": "Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred."
      }
    ],
    "target": {
      "record_id": "urn:uuid:a572a20b-6642-435c-a2d7-747a12e7faa6",
      "version": 1,
      "content_hash": "sha256:27d669f8f51cd2afe17f69d5b1ff3649dbce8db90c77d8de8422867a62c94de5"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "220000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": {
    "id": "darboux-intermediate-value-property-derivatives-1875",
    "target": {
      "record_id": "urn:uuid:a572a20b-6642-435c-a2d7-747a12e7faa6",
      "version": 1,
      "content_hash": "sha256:27d669f8f51cd2afe17f69d5b1ff3649dbce8db90c77d8de8422867a62c94de5"
    },
    "sections": [
      {
        "title": "Can a slope skip a value?",
        "text": "For a smooth-looking curve, it seems natural that slopes change continuously. But differentiability of a function does not force continuity of its derivative. Darboux identifies a different universal constraint: when the finite derivative exists everywhere in an interval, it cannot skip an intermediate value. If its slope is one at one location and three at another, the slope must be two somewhere between them. This remains true even when the derivative is discontinuous."
      },
      {
        "title": "Subtract the slope you want to find",
        "text": "Choose a desired intermediate slope m and subtract the straight line m times x from the original function. At the two selected endpoints, the new derivative has opposite signs. In the appropriate orientation the new function begins by increasing and ends by decreasing, so its maximum occurs at an interior point. At that maximum the derivative is zero. Undoing the subtraction shows that the original derivative equals m there. The argument turns a slope question into an extremum question."
      },
      {
        "title": "Oscillation differs from a jump",
        "text": "Define a classroom function as x squared times sin of one divided by x away from zero, and set its value at zero to zero. Its derivative at zero is zero, while away from zero the derivative contains a cosine term that keeps oscillating. The derivative is discontinuous at zero, yet it does not create a clean jump that skips all intermediate values. The example shows why discontinuous is a broader category than having a simple step discontinuity."
      },
      {
        "title": "Do not reverse the implication",
        "text": "Every derivative in the stated scope has the intermediate-value property. It does not follow that every function with that property is a derivative. Nor may the hypothesis be weakened silently to a derivative existing only almost everywhere. A step function illustrates the importance of the condition: it can jump from one level to another without taking intermediate levels, so it cannot be an everywhere-existing finite derivative on that interval. The theorem provides a necessary condition, not a complete characterization."
      },
      {
        "title": "An attainable source-and-example check",
        "text": "Ask your agent to differentiate the oscillating example away from zero and compute the derivative at zero from its definition. It should explain why a graph sampled at finitely many points cannot establish continuity or the intermediate-value property by itself. Then have it identify the exact derivative hypothesis in the source. The manuscript signature is from1874and the journal publication from1875; those dates serve different roles. A concise checked explanation is a useful contribution without claiming a new theorem."
      }
    ],
    "references": [
      {
        "title": "Mémoire sur les fonctions discontinues — original 1875 journal scan, especially §IX pp.109–112",
        "url": "https://www.numdam.org/item/10.24033/asens.122.pdf",
        "locator": "Original SectionIXpp109–112,oscillating derivative example and subtract-a-line proof;title/history introductory pages."
      }
    ]
  },
  "reading_status": "EXPLANATORY_GUIDE"
}