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    "title": "Baire: pointwise-limit function classes and the first-class characterization",
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        "description": "Pointwise convergence need not be uniform and does not make the limit continuous. Historical punctually discontinuous means having continuity points densely in the relevant domain or perfect subset, not being discontinuous at every point; the condition concerns the restricted function. Exact-class conventions exclude lower classes, unlike the modern cumulative at-most-class convention. The 1899 chapter does not prove nonemptiness or a full characterization of every higher class; printed page 71 explicitly leaves that programme open. The 1900 paper credits Lebesgue’s reduction of 27 March 1899 for the multivariable extension, so its entire extension is not assigned exclusively to Baire. No arbitrary-domain, vector-valued or transfinite-uniform-convergence theorem is inferred.",
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      "statement": "# Baire: pointwise-limit function classes and the first-class characterization\n\nA sequence of continuous functions can settle point by point into a discontinuous limit. Baire classified functions by repeated pointwise limits and characterized the first class on a real interval through continuity of the restricted function on every perfect subset. His thesis distinguishes successive classes and gives a rational/irrational class-two example. A 1900 paper supplies a shorter sufficiency proof on a finite-dimensional closed cube, while explicitly acknowledging Lebesgue’s reduction of several variables to one.\n\n## Scientific scope\n\nPointwise convergence need not be uniform and does not make the limit continuous. Historical punctually discontinuous means having continuity points densely in the relevant domain or perfect subset, not being discontinuous at every point; the condition concerns the restricted function. Exact-class conventions exclude lower classes, unlike the modern cumulative at-most-class convention. The 1899 chapter does not prove nonemptiness or a full characterization of every higher class; printed page 71 explicitly leaves that programme open. The 1900 paper credits Lebesgue’s reduction of 27 March 1899 for the multivariable extension, so its entire extension is not assigned exclusively to Baire. No arbitrary-domain, vector-valued or transfinite-uniform-convergence theorem is inferred.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nRené Baire\n\nSource-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.\n\n\n\n## Original sources\n\n- [Sur les fonctions de variables réelles (original Milan thesis edition,1899)](https://archive.org/download/surlesfonctions00bairgoog/surlesfonctions00bairgoog.pdf)\n- [Nouvelle démonstration d’un théorème sur les fonctions discontinues (BulletinSMF28,173–179)](https://www.numdam.org/item/10.24033/bsmf.631.pdf)\n",
      "summary": "A sequence of continuous functions can settle point by point into a discontinuous limit. Baire classified functions by repeated pointwise limits and characterized the first class on a real interval through continuity of the restricted function on every perfect subset. His thesis distinguishes successive classes and gives a rational/irrational class-two example. A 1900 paper supplies a shorter sufficiency proof on a finite-dimensional closed cube, while explicitly acknowledging Lebesgue’s reduction of several variables to one.",
      "title": "Baire: pointwise-limit function classes and the first-class characterization"
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    "what_changed": "A sequence of continuous functions can settle point by point into a discontinuous limit. Baire classified functions by repeated pointwise limits and characterized the first class on a real interval through continuity of the restricted function on every perfect subset. His thesis distinguishes successive classes and gives a rational/irrational class-two example. A 1900 paper supplies a shorter sufficiency proof on a finite-dimensional closed cube, while explicitly acknowledging Lebesgue’s reduction of several variables to one.",
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