In 1890 René completed the advanced classes at the Lycée Lakanal and entered the special mathematics section of the Lycée Henri IV. After completing one year of preparation at this Lycée, he passed the entrance examinations for both the École Polytechnique and the École Normale Supérieure. He chose the latter as the place to study. Costabel writes in :-
... during his three years there [he] attracted attention by his intellectual maturity. He was a quiet young man who kept to himself and was profoundly introspective. During this period he was found to be in delicate health.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. While he was a student he assisted with the editing of Poincaré's lectures, which he attended in 1894, on the propagation of heat. 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. He was asked to prove the continuity of the exponential function but when he was in the middle of the proof he realised that :-
... his demonstration of continuity, which he had learnt at the Lycée Henri IV, was purely an artifice, since it did not refer sufficiently to the definition of the function.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. However, the immediate result of his passing his agregation was that he obtained his first post as a professor at a lycée. An appointment in Bar-le-Duc gave him reasonable financial security but he was unhappy that living in Bar-le-Duc meant that he had no opportunity for close contacts with university life.
At the Lycée in Bar-le-Duc Baire worked on the theory of functions and the concept of a limit. Around this time he discovered conditions under which a function is a limit of a sequence of continuous functions. Shortly after this Baire set up his classification of functions. Class 1 functions were those functions which were the limit of a sequence of continuous functions. Class 2 functions were those functions which were the limit of a sequence of Class 1 functions, while Class 3 functions were those functions which were the limit of a sequence of Class 2 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. He was examined on 24 March 1899 by a board consisting of Darboux, Appell and Émile Picard, and they awarded him the doctorate. However :-
The few objections, which Baire fully appreciated, proved that he had embarked on a new road and would not find it easy to convince his listeners.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. He continued to teach in lycées (he taught in Troyes, Bar-le-duc and Nancy) but was not happy teaching at this low level. In 1901 Baire was appointed to the University of Montpellier as a "Maitre des conferences". This post saw him preparing students for the "agregation" examination, a position he enjoyed much more than teaching in lycées. While at Montpellier he wrote a paper on irrational numbers and limits.
In 1904 he was awarded a Peccot Foundation Fellowship which was to allow young school teachers to spend a semester in a university developing their skills. 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. He was appointed to a university post in 1905 when he joined the Faculty of Science at Dijon. In 1907 he was promoted to Professor of Analysis at Dijon.
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. The bad spells became more frequent, immobilising him for long periods. Apart from problems with his oesophagus that had plagued him since his youth, he developed a kind of psychological disorder which, using his own description, "debilitated" him occasionally. Apparently he eventually became unable to undertake work which required him to concentrate, and research in mathematics became impossible at these times. Between 1909 and 1914 he continued trying to undertake his teaching duties, but this became more and more difficult. Then near the beginning of 1914 he requested leave so that he might try to recover his health.
Baire went first to Alésia, then he went to Lausanne. It was while he was in Lausanne that World War I began and he was not able to return to France. He spent the war years from 1914 until 1918 in Lausanne in quite difficult financial circumstances.
It is interesting to consider the various causes suggested by his contemporaries to account for his illness. Some suggested that the cause for his problems lay in intellectual overexertion in his student days. His close family, and others close to him, blamed his illness on his deep feelings of frustration that his achievements were not being recognised by the academic authorities. 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. He first fell out with Lebesgue in 1904, when he taught his course at the Collège de France, over who had the most right to teach such a course. 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 letters written to Baire by de la Vallée Poussin, and reproduced in , give an idea of the reasons for their arguments which seem to centre round the fact that de la Vallée Poussin had classified by order of importance mathematical discoveries of Lebesgue and Baire.
While on the topic of letters, we should remark that  contains fifty letters written by Baire to Émile Borel. The first five are written during 1898 beginning during the time that Baire was in Italy. There is a gap from the fifth letter, dated 22 May 1898 to the sixth dated 4 February 4 1902. The gap is explained by Baire's first serious illness over the period he taught at Bar-le-Duc. In the letters reproduced in , Baire writes in great detail about his research ideas, including the Baire classification of functions, sets of first and second category, and semicontinuity. In the letters he also discusses Cantor's set theory and the foundations of mathematics.
It appears that it was not only Baire's family who felt he had been hard done by, for after 1918 many in the mathematical community seemed to be trying to make amends for his lack of recognition. 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. He retired in 1925 and spent his last years in the solitude of hotel rooms on the shores of the Lake of Leman. Although he received a reasonable pension, inflation over these years soon meant that he ended his life with financial difficulties similar to those of his youth.
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 Ⓣ (1905) and Leçons sur les théories générales de l'analyse Ⓣ, 2 Vols. (1907-8). 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. He wrote in his doctoral thesis:-
Generally speaking, in the framework of ideas that here concern us, every problem in the theory of functions leads to certain questions in the theory of sets, and it is to the degree that these latter questions are resolved, that it is possible to solve the given problem more or less completely.When his health was good, the quality of his lectures received rather differing opinions from his students. Some described his lectures as very clear, but others claimed that what he taught was so difficult that it was beyond human ability to understand. Baire, aware of these comments, wrote:-
... but look at Denjoy - he understood it, hence it must not be so difficult ...Denjoy, who was Baire's most famous student, certainly understood Baire's ideas and developed them in his own work. He wrote that Baire was:-
.. not an agreeable character ... [and] not a person of enormous culture ...[but] constantly tormented due to the fatigue of his brain.On the other hand Denjoy described him as:-
... an excellent person.Another of his students, Reault, wrote much more positively describing Baire as having:-
... paternal concern ... [He had] high intellectual qualities ... a penetrating mathematical mind [with] the extent and depth of his knowledge ... [and] the greatness of his character.
Article by: J J O'Connor and E F Robertson