**Denjoy** was a French mathematician who did outstanding work on the functions of a real variable.

- Arnaud was born and attended a secondary school in Auch which is the capital town of the Gers region in southwestern France.
- After attending the Lycée d'Auch, Arnaud completed his schooling at the lycée in Montpellier, which is a city close to the Mediterranean coast in southern France.
- At the lycée in Montpellier, Denjoy showed a great talent for arts subjects and his rhetoric teacher gave him the following advice: "Denjoy, you have a lot of ideas for your age, don't do mathematics, you can become someone." Denjoy did not follow this advice and he took the special mathematics class which was designed to train pupils to sit the entrance examinations for the École Polytechnique and the École Normale Supérieure.
- He then had a stroke of luck, for one of the boys who had won a place pulled out and Denjoy was able to enter.
- In 1902 Denjoy entered the École Normale Supérieure where he soon showed his relatively poor performance in the entrance examination did not match his remarkable abilities.
- These great mathematicians gave Denjoy a strong background in complex function theory, continued fractions and differential equations and set him on the road to his great discoveries.
- Denjoy enjoyed the highest success during his undergraduate years, being the top student in his class when obtaining the Agrégé de mathématiques in 1905.
- Also in 1905 the book "Leçons sur les fonctions discontinués: professées au Collège de France" Ⓣ(Lessons on discontinued functions: taught at the Collège de France) by René Baire and Denjoy was published.
- On the title page Baire appears as a lecturer at the Faculty of Science at Montpellier, while Denjoy as a student at the École Normale Supérieure.
- Denjoy's success as an undergraduate was translated into the winning of a prestigious fellowship, the Fondation Thiers fellowship, which supported him for three years during which he worked on his dissertation Sur les produits canoniques d'ordre infiniⓉ(On canonical products of infinite order) which he submitted in 1909.
- Denjoy's 1909 dissertation, although not considered by him as among his greatest achievements when he looked back on his career in 1934, is now considered to contain some remarkable contributions.
- From Paris, Denjoy moved back to Montpellier in 1909 when he was appointed "Maitre des conferences" at the University of Montpellier on 14 October.
- Baire had left Montpellier before Denjoy took up this appointment, but he was an important figure in the development of Denjoy's mathematics.
- Denjoy was one of a small number of people who appreciated Baire's innovative ideas when he first produced them.
- but look at Denjoy - he understood it, hence it must not be so difficult ...
- Denjoy worked on functions of a real variable in the same areas as Borel, Baire and Lebesgue.
- These four volumes were an expanded version of work which had appeared in a series of papers by Denjoy beginning in 1920.
- Included in these papers was his introduction of the Denjoy index for the points of a perfect set.
- Similarly Denjoy's theorem on quasi-analytic functions has been the foundation of studies by Mandelbrojt and has proved important in the development of large areas of current research.
- In 1946 Denjoy published L'Énumération Transfinie.
- Without minimising the achievements of the axiomatic approach, Denjoy pleads for a return to descriptive notions in the theory of sets of points, and for an honest admission of what is actually 'thought' behind purely formal phraseology.
- Denjoy gave the lecture Le Mécanisme des Opérations Mentales chez les Mathématiciens Ⓣ(The Mechanism of Mental Operations in Mathematicians) on 25 November 1947 at the Romanian Institute of Science and Technology.
- In 1956 Denjoy published Un Demi-siècle (1907-1956) de Notes Communiquées aux Académies.
- Let us end by saying something of Denjoy's character.
- Finally the strikers, after much urging, agreed to return to the lecture hall but on one condition: that Denjoy should cease lecturing in English and instead lecture in French.
- We learn much of Denjoy's interests by reading his account of his trip to Rome in 1908 to attend the International Congress of Mathematicians.
- Denjoy was not a man lacking interests outside mathematics: on the contrary he was fascinated by topics such as philosophy, psychology, and social studies.
- For his outstanding contributions to the theory of functions of a real variable, Denjoy received many honours.

Born 5 January 1884, Auch, Gers, France. Died 21 January 1974, Paris, France.

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