**René Baire** worked on the theory of functions and the concept of a limit.

- René grew up in Paris at the time when the Eiffel tower was being constructed.
- In 1890 René completed the advanced classes at the Lycée Lakanal and entered the special mathematics section of the Lycée Henri IV.
- At the École Normale Supérieure Baire attended lectures by Jules Tannery and Goursat and, in addition, he attended lectures by Hermite, Émile Picard and Poincaré at the Sorbonne.
- Having received his licentiate Baire proceeded toward his "agregation" but, although he was the best student in the written parts of this examination, he was only third overall after the oral examination.
- His poor performance in the oral is worth describing in more detail since it was to have a great affect on the future direction of Baire's research.
- The examiners were hard on Baire and he was extremely disappointed with the outcome, but he then determined to examine again his analysis course while researching into the concept of continuity of a general function.
- At the Lycée in Bar-le-Duc Baire worked on the theory of functions and the concept of a limit.
- Shortly after this Baire set up his classification of functions.
- Baire was awarded a scholarship to allow him to continue his studies in Italy and there he met and established a close friendship with Volterra.
- While he worked in the lycée, Baire wrote a doctoral thesis on discontinuous functions.
- Even before presenting his thesis Baire had suffered from poor health and, after the award of his doctorate, he was only able to contribute to mathematics for a few short spells.
- In 1901 Baire was appointed to the University of Montpellier as a "Maitre des conferences".
- Baire spent the semester at the Collège de France where he lectured on the subject of his thesis and had the lectures published the next year.
- Baire returned to Montpellier where he suffered the first severe attack of illness but after a while the worst of the attack passed and he was able to work again.
- Baire's health had never been good since he was young but from the time he was at the Lycée at Bar-le-Duc it began to deteriorate to the stage that it prevented him from working.
- Baire went first to Alésia, then he went to Lausanne.
- Baire felt that he deserved a professorship in Paris and failing to achieve this, it was suggested, caused him depression and hence his ill health.
- Certainly Baire felt that men such as Lebesgue, who was younger than Baire, had been unfairly preferred to him.
- Their rivalry turned into a more serious argument later in Baire's life.
- Baire also fell out with de la Vallée Poussin which may be surprising to those who know that Baire's ideas entered the mainstream of mathematics through de la Vallée Poussin's well-known treatise.
- The first five are written during 1898 beginning during the time that Baire was in Italy.
- The gap is explained by Baire's first serious illness over the period he taught at Bar-le-Duc.
- In 1918 some suggested that a chair at the Collège de France, which he undoubtedly deserved, would lift Baire's depression, helping him to regain his intellectual vigour, but apparently these suggestions never materialised.
- Unable to resume his duties, Baire lived on the shores of Lake Geneva and he was there when he received the Chevalier de la Legion d'Honneur and in 1922 when he was elected to the Académie des Sciences.
- Despite being unable to work for long periods, Baire wrote a number of important analysis books including Théorie des nombres irrationels, des limites et de la continuité Ⓣ(Theory of irrational numbers, limits and continuity) (1905) and Leçons sur les théories générales de l'analyse Ⓣ(Lessons on general theories of analysis), 2 Vols.
- Baire made a decisive step in moving away from the intuitive idea of functions and continuity and he saw clearly that a theory of infinite sets was fundamental for rigorous real analysis.
- Denjoy, who was Baire's most famous student, certainly understood Baire's ideas and developed them in his own work.

Born 21 January 1874, Paris, France. Died 5 July 1932, Chambéry, France.

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Bourbaki

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive