# Arnaud Denjoy

### Quick Info

Auch, Gers, France

Paris, France

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

### Biography

**Arnaud Denjoy**'s father, Jean Denjoy, was a wine merchant in Perpignan, which is in southern France about 30 km north of the border with Spain. For a picture of the yard of 'Jean Denjoy Wholesale Wines, Perpignan', see THIS LINK.

Arnaud's mother was Camille Henriette Maria Jayez; she came from Catalonia. Jean and Camille were married in Pavie in 1880. Arnaud had an older sister Marie Antoinette Dejoy, born 16 April 1881. Marie Antoinette married Auguste Henri Marie Pagnon (1878-1913) on 30 November 1901 in Perpignan.

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. He remained associated with these regions throughout his life. 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. Denjoy [8]:-

... had, in the special mathematics class of Montpellier, a professor of mathematics whom "the demands of logic made furious"; so, with his fellow students, he had to learn from old courses and from books.After this somewhat deficient preparation he did not rank highly enough to enter the École Normale, being the best student to miss out on a place. 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. He studied under Émile Borel, Paul Painlevé and Émile Picard. 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. Remarkably, his first paper,

*Sur quelques propriétés des fonctions de variables réelles*Ⓣ, appeared in the year he graduated. This is not a short note but rather it is a substantial 17-page work published in the

*Bulletin*of the French Mathematical Society. Also in 1905 the book

*Leçons sur les fonctions discontinués: professées au 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. Baire writes in the Preface:-

The publication of these lessons was considerably facilitated for me by the assistance of one of my students, M Denjoy, who took on the task of writing it. I give him my thanks.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*Ⓣ which he submitted in 1909. He wrote in 1934 [2]:-

... during the previous year 1908, I had taken sides in a controversy on the question, formerly raised by Painlevé in his Thesis, namely the appearance of the functions of a complex variable, uniform throughout the plane, and provided with a completely discontinuous perfect set of singularities. I gave simple examples of a function remaining bounded on a set of finite positive length, then of a continuous function on a set of infinite length and zero area. On this occasion my attention and that of other Frenchmen turned to the properties of perfect planar sets which are totally discontinuous.He had published the paper

*Sur les fonctions entieres de genre fini*Ⓣ in 1907 and then three papers in 1909:

*Sur les fonctions analytiques uniformesa singularités discontinues*Ⓣ;

*Sur les fonctions analytiques uniformes qui restent continues sur un ensemble parfait discontinu de singularités*Ⓣ; and

*Sur les singularités discontinues des fonctions analytiques uniformes*Ⓣ.

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. It studies the asymptotic behaviour of integral functions of finite order, Weierstrass products of integral functions and the boundary behaviour of conformal representations.

From Paris, Denjoy moved back to Montpellier in 1909 when he was appointed "Maitre des conferences" at the University of Montpellier on 14 October. This post saw him preparing students for the "agrégation" examination and it was the same post in the same university to which Baire had been appointed in 1901. 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. For example when Baire was told that what he taught was so difficult that it was beyond human ability to understand it, he wrote:-

... but look at Denjoy - he understood it, hence it must not be so difficult ...Writing about his earlier work in 1934 Denjoy wrote [2]:-

My research was soon to take on a more personal and independent character. At the end of 1909 I ended my studies on perfect sets which were totally discontinuous planes and uniform analytical functions of which they constituted the only singularities. Nevertheless the theory of uniform functions of a complex variable pushed me towards the topology of the plane and particularly the study of the borders of regions where a function remains holomorphic (a region which is connected and formed of interior points). At this time the subject was dominated by the problem of giving a correct demonstration of Jordan's famous theorem on the division of the plane into two regions by any homeomorphic line at a circumference. Many childishly intuitive reasonings claimed to resolve the question. One could think of the grid of the plan, as Jordan had given the example for his measurement of sets. I observed that on the one hand the torus, on the other hand the surface with only one side, lent themselves to the grid and that any demonstration not invoking characters of the plane excluding the application of reasoning to the two varieties just cited could not be considered satisfactory.Denjoy taught at Montpellier until the start of World War I in 1914. He suffered from poor eyesight so he was not fit for active military service during the war but he was mobilised in the auxiliary service. He wrote [2]:-

In 1914 came the war, with four years of fighting, where science played a decisive role. As soon as the new map of Europe was adopted, as early as 1920 scientific studies, among other things mathematics, took off.He appealed against his auxiliary military service and was allowed to take up a professorship at Utrecht beginning on 1 October 1917. He had published some major works in the preceding couple of years which had shown that he was a world-leading researcher. For example there was the 88-page paper

*Sur les fonctions dérivées sommables*Ⓣ and the 136-page paper

*Mémoire sur les nombres dérivés des fonctions continues*Ⓣ in 1915, the 96-page paper

*Mémoire sur la totalisation des nombres dérivées non sommables*Ⓣ in 1916 and the 58-page paper

*Mémoire sur la totalisation des nombres dérivées non sommables (suite)*Ⓣ in 1917. Henri Cartan writes [8]:-

At the height of the submarine war, [Denjoy] joined his new post by sea, via England, and his boat was torpedoed twice. He then spent five of the most successful years in Utrecht, the memory of which he often enjoyed. It was the era of long bicycle rides along the canals and of collaboration with Julius Wolff (1882-1945).He was called to the Faculty of Science in Paris beginning in 1922, first as a lecturer, then as an assistant professor, being promoted to full professor in 1931, a post he held until he retired in 1955. Not long after his appointment at the University of Paris, he married Thérèse-Marie Chevresson on 15 June 1923. Marie-Thérèse, the daughter of the tax collector Charles Marie Chevresson and Melanie Françoise Berthe Becker, had been born in Charleville-Mézières (Ardennes) on 26 April 1892. They had three children, all sons: Jean Fabrice Denjoy (born 1 January 1925), who became a doctor of medicine and married Caroline-Adams Byrd; René Denjoy (born 17 September 1927), who also became a doctor of medicine and surgeon; and Bernard Denjoy (born 23 June 1926), who became an architect and married Anne-Marie Roederer. We note that Denjoy asked Nikolai Luzin to he godfather to René Denjoy. Luzin said to Denjoy:-

I heartily thank you for the honour and friendship you show me by choosing me as the godfather for your little René. ... As you well know, christening has for me a profound meaning.Let us quote from Denjoy himself, writing in 1934, about his work (see [2]):-

My luck was twofold: on the one hand, that the solutions to these problems waited for my coming, afterwards, that they appeared to me. Far be it from me to reduce the role of those who preceded me in this long ascent to the targeted summits. If the works of my predecessors, and those of M Lebesgue which dominate them all, had not previously been smoothed out and marked out by hard and laborious stages, my starting base would have been undoubtedly too far from the goal for me to be able to reach that - by my only effort.Denjoy worked on functions of a real variable in the same areas as Borel, Baire and Lebesgue. He combined topological and metrical methods to attack problems of real analysis. In 1934 he wrote that his greatest achievements had been the integration of derivatives, the computation of the coefficients of a converging trigonometric series, a theorem on quasi-analytic functions, and differential equations on a torus.

However, the last step still to be crossed, and at the foot of which the previous attempts had stopped, was also not without serious difficulties to overcome. Now that the ridge is dominated, that we can measure its elevation, perhaps we will consider that the summit was still far from the extreme point previously reached.

The solutions that I have provided do not fit into the standard of classical mathematics. They have therefore been unequally appreciated.

All the mathematicians who embarked on the new paths of the theory of the functions of real variables, and who tested the conviction of penetrating into a world of truths illuminating a vast field of facts, all suffered from the feelings of indifference, of hostility, aversion to which their discoveries met with by scholars loyal to traditional studies. Everyone complained bitterly about it. I will not escape the common rule, although, in the face of the enormous development of modern Analysis, and the fruitful invasion of new ideas into the old fields which seem to be the most reserved, the prejudices of thirty or forty years ago are today much attenuated, it should fairly be recognized.

What I would criticise for these unfriendly provisions with regard to the new disciplines is the spirit, insufficiently scientific in my opinion, from which they proceed. For me, mathematics requires the same objectivity as an experimental science.

The second of these topics, computation of the coefficients of a converging trigonometric series, was the subject of a four volume work

*Leçons sur le calcul des coefficients d'une série trigonométrique*Ⓣ which appeared between 1941 and 1949. The volumes were subtitled: 1.

*La différentiation seconde mixte et son application aux séries trigonométriques*Ⓣ; 2.

*Métrique et topologie d'ensembles parfaits et de fonctions*Ⓣ; 3.

*Détermination d'une fonction continue par ses nombres dérivés seconds généralisés extrêmes finis*Ⓣ; and 4.

*Les totalisations, solution du problème de Fourier*Ⓣ. 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. Also in these papers is his introduction of the second symmetric derivative of a function. This work is studied in detail in [6] where Bullen makes the contents of Denjoy's work, which is not easy to read, more accessible to modern analysts.

Although considered by Denjoy as one of his most important pieces of work in his 1934 review, Choquet writes in [9] that these papers may be:-

... considered more feats of intellectual strength than sources of practical applications.However, Choquet describes the four volume work

*Leçons sur le calcul des coefficients d'une série trigonométrique*Ⓣ which contains the famous Denjoy integral, as [9]:-

... an explosion of beautiful theorems and examples.Choquet, very fairly, suggests that Denjoy's work on differential equations on a torus, not nearly so highly rated by Denjoy himself, is one of his most influential pieces of work and has [9]:-

... grown into a vast field involving dynamical systems.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. Livre I. La Notion de Rang*Ⓣ. Alonzo Church writes in a review [11]:-

Addressed to mathematical analysts, this is a treatise on transfinite ordering, well-ordered series, and ordinal numbers, with attention primarily to the second number class. In an otherwise excellent work, the treatment of the axiom of choice, and of Zermelo's theorem that every class can be well ordered, is without value, because the author mistakenly identifies the axiom of choice with the proposition that every non-empty class has a unit subclass. A second volume was already written in 1942 but publication of it must still be postponed because of difficulties created by the war. The table of contents published in the present volume indicates that two appendices will contain more about Zermelo's theorem, and some discussion of such matters as the Richard paradox, the relationship between effective definition and existence.

*L'Énumération Transfinie*Ⓣ became a four volume work, the other three volumes being entitled: Vol. II/l.

*Les permutations spéciales*Ⓣ (1952); Vol. II/2.

*Les suites canoniques*Ⓣ (1952); Vol. III.

*Etudes complémentaires sur l'ordination*Ⓣ (1954); and Vol. IV.

*Notes sur les sujets controversés*Ⓣ (1954). Rosalind Cecilia Hildegard Young, the eldest daughter of Grace Chisholm Young and William Young, wrote [29]:-

In 1942, the work purported to be a 'mise au point' in the standing controversy on transfinite numbers. In 1954, it has to contend, besides, with the formidable inroads of axiomatism, described legitimately as scholasticism in the mathematical field. ... 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*Ⓣ 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. I La variable complexe*Ⓣ. Walter Hayman writes [18]:-

This is a very stimulating book and evokes admiration for the author of so many fertile ideas in the theory of functions, varying over integral functions, quasi analytic functions, discontinuous perfect sets of singularities, Fuchsian groups, conformal mappings, and so on. Many of the ideas are merely sketched. This collection will be an inspiration to the young researcher in function theory, and ought to be required reading for him. He should, however, take care to scan the literature for the possibility that others also have worked in the same field, since the author's references to work other than his own are few and far between.Gustave Choquet writes [9]:-

I heard him repeatedly regret not having read enough and thus having lost opportunities to broaden his mathematical horizon. It is true that he read very few mathematical publications, and almost exclusively the reprints that were sent to him. But the variety of his work in the most productive part of his life makes one doubt the correctness of this self-criticism; in fact his exceptional geometric intuition combined with his great power of concentration probably compensated for the rarity of his contacts with mathematical literature, and these two factors contributed to preserve his originality.Let us end by saying something of Denjoy's character. He was a quiet man, who liked to carry out his mathematical research in the peace of his own home. For leisure he enjoyed being in the countryside, particularly walking and cycling in wooded country. A relatively poor lecturer he was, nevertheless, a fine writer and an entertaining man with whom to hold a conversation. The following anecdote may not be true but it almost certainly contains a grain of truth. It concerns a visit Denjoy made to Columbia University on the early 1950s:-

At one point, Denjoy was giving a lecture series to an audience that began with a substantial number of subscribers but quickly frittered down to just three students. After a few more lectures, these three decided to go on strike, claiming that their situation was untenable. The entire department was in an uproar. 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. $$

We note that he also attended the International Congress of Mathematicians in 1920 when he gave the lecture

*Sur une classe d'ensembles parfaits en relation avec les fonctions admettant une dérivée seconde généralisée*Ⓣ, the Toronto Congress in 1924, and the Bologna Congress in 1928 when his wife accompanied him. In 1950 he attended the 1950 Congress in Cambridge, Massachusetts, giving the lecture

*Les permutations clivées*Ⓣ in the Analysis Section on Saturday 2 September. He was also the Honorary Chairman of the Conference in Analysis held on Friday 1 September. In 1954 he attended the Congress in Amsterdam, accompanied by his son Bernard Denjoy, and Bernard's wife. In 1958 he attended the Congress in Cambridge, England and gave the lecture

*Approximation des nombres réels par ceux d'un corps fourni par un groupe fuchsien*Ⓣ. At the Stockholm Congress in 1962 he gave the lecture

*Les équations différentielles périodiques*Ⓣ. Also, at the age of 86, he attended the 1970 Congress in Nice, France.

Denjoy was not a man lacking interests outside mathematics: on the contrary he was fascinated by topics such as philosophy, psychology, and social studies. He approached such topics from his position as an atheist and an active participant in socialist politics. In particular he was an active member of the Radical Party which was headed by Édouard Herriot. Although Denjoy did not aspire to the political career of Herriot, who served in nine different cabinets and was premier of France three times, Denjoy's involvement with the Radical Party led to him serving as a town councillor for Montpellier in 1912, and as county councillor for Gers from 1920. He served in this capacity for twenty years.

For his outstanding contributions to the theory of functions of a real variable, Denjoy received many honours. He was elected President of the French Mathematical Society in 1931. As well as election to the Geometry Section of the Paris Académie des Sciences on 15 June 1942, succeeding Henri Lebesgue whose work he had extended, he was vice-president in 1961 and president in 1962. Henri Cartan writes [7]:-

Arnaud Denjoy presided over our Academy in 1962. No one more than he was convinced of the need to reform it, so that in all scientific disciplines the most remarkable men of our country would be called upon to put themselves at the service of the Academy when "They are still in the prime of life and in full activity." As President, he had submitted certain suggestions to the Academy, and it was a great disappointment to him to note that they had not resulted in any concrete decision. Hopefully the ideas he sowed will one day find the ground that will allow them to germinate and flourish!He was also elected to the Royal Dutch Academy of Sciences, the Polish Academy of Sciences, and the Academy of Liege. He was elected an honorary member of the Belgium Mathematical Society; an honorary member of the Moscow Mathematical Society; an honorary member of the Romanian Academy of Sciences (1938); an honorary member of the Boston Academy of Arts and Sciences (1939); an member of the Russian Academy of Sciences; and an honorary member of the Lima Academy of Sciences. He was also honoured by being elected as vice-president of the International Mathematical Union in 1954. He was awarded the Saintour Prize (1925); the Poncelet Prize (1930); the Petit d'Ormoy Prize (1933); Albert 1 of Monaco Prize (1938); and the Lomonosov gold medal (1970) for outstanding achievements in mathematics.

He died aged 90 after a fall in his home.

### References (show)

- G Choquet, Biography in
*Dictionary of Scientific Biography*(New York 1970-1990). See THIS LINK. - G Choquet,
*Arnaud Denjoy : évocation de l'homme et de l'oeuvre, Astérisque***28-29**(Paris, 1975). - F Le Lionnais,
*Great Currents of Mathematical Thought: Mathematics: Concepts and Development*(Courier Corporation, 2004). - Arnaud Denjoy. Bibliographie, Arnaud Denjoy: évocation de l'homme et de l'oeuvre,
*Astérisque***28-29**(1975), 7-13. - R Blanché, Review: Hommes, formes et le nombre, by Arnaud Denjoy,
*Revue Philosophique de la France et de l'Étranger***156**(1966), 506. - P S Bullen, Denjoy's index and porosity,
*Real Anal. Exchange***10**(1) (1984/85), 85-144. - H Cartan, Notice nécrologique sur Arnaud Denjoy, Arnaud Denjoy: évocation de l'homme et de l'oeuvre,
*Astérisque***28-29**(1975), 14-18. - H Cartan, Notice nécrologique sur Arnaud Denjoy, membre de la section de géométrie,
*Comptes rendus de l'Académie des Sciences Paris Vie Académique***279**(1974), 49-53. - G Choquet, Avant-Propos, Arnaud Denjoy: évocation de l'homme et de l'oeuvre,
*Astérisque***28-29**(1975), 3-6. - C Christophe and E Telkès, Arnaud Denjoy, in
*Les Professeurs de la faculté des sciences de Paris, 1901-1939, Dictionnaire biographique (1901-1939)*(Institut national de recherche pédagogique, Paris, 1989), 107-109. - A Church, Review: L'Énumération Transfinie. Livre I. La Notion de Rang, by Arnaud Denjoy,
*The Journal of Symbolic Logic***13**(3) (1948), 144. - A Denjoy, Address by Professor A Denjoy, laureate of the M V Lomonosov Gold Medal (Russian),
*Vestnik. Akad. Nauk SSSR*(5) (1971), 57-64. - A Denjoy, Mon oeuvre mathématique. Sa genèse et sa philosophie, Arnaud Denjoy: évocation de l'homme et de l'oeuvre,
- A Denjoy, Le mécanisme des opérations mentales chez les mathématiciens, Arnaud Denjoy: évocation de l'homme et de l'oeuvre,
*Astérisque***28-29**(1975), 43-54. - A Denjoy, Rome 1908, Arnaud Denjoy: évocation de l'homme et de l'oeuvre,
*Astérisque***28-29**(1975), 55-64. - P Dugac, Nicolas Lusin: lettres à Arnaud Denjoy,
*Arch. Internat. Hist. Sci.***27**(10) (1977), 179-206. - S Dimiev and B Penkov, Arnaud Denjoy (1881-1974) (Bulgarian),
*Fiz.-Mat. Spis. Bulgar. Akad. Nauk.***17**(50) (1974), 152-156. - W K Hayman, Review: Un Demi-siècle (1907-1956) de Notes Communiquées aux Académies. I La variable complexe, by Arnaud Denjoy,
*The Mathematical Gazette***43**(343) (1959), 72. - B Jessen, Review: Introduction à la théorie des fonctions de variables réelles, I-II, by Arnaud Denjoy,
*Matematisk Tidsskrift. B, tematisk tidsskrift. B*(1938), 57. - C H Langford, Review: La Part de l'Empirisme dans la Logique Mathématique by Arnaud Denjoy,
*The Journal of Symbolic Logic***3**(1) (1938), 56. - P Lévy, Review: Hommes, formes et le nombre, by Arnaud Denjoy,
*Revue de Métaphysique et de Morale***71**(1) (1966), 108-110. - M Luzin, A letter to Arnaud Denjoy in 1926 (Polish),
*Wiadom. Mat.***25**(1) (1983), 65-68. - F A Medvedev (trans.), N N Luzin's letters to A Denjoy,
*Istor.-Mat. Issled. No.***23**(1978), 314-348; 359. - G H Müller, Review: L'énumération transfinie. Vol. I. La notion de rang (1946). Vol. II/1. Les permutations spéciales (1952). Vol. II/2. Les suites canoniques (1952). Vol. III. Etudes complémentaires sur l'ordination (1954). Vol. IV. Notes sur les sujets controversés (1954) by Arnaud Denjoy,
*Dialectica***8**(3) (1954), 270-273. - K Petrova (trans.), Arnaud Denjoy (1884-1974): on the occasion of the 100th anniversary of his birth (Bulgarian),
*Fiz.-Mat. Spis. Bulgar. Akad. Nauk.***26**(59) (2) (1984), 202-208. - C N S, Review: Introduction à la théorie des fonctions de variables réelles, I-II, by Arnaud Denjoy,
*Current Science***6**(3) (1937), 122. - J B Rosser, Review: L'Enumeration Transfinie, by Arnaud Denjoy,
*The Journal of Symbolic Logic***21**(1) (1956), 95-96. - A Virieux-Reymond, Review: L'énumération transfinie. Fascicule IV: Notes sur les questions controversées, by Arnaud Denjoy,
*Revue de Théologie et de Philosophie*(3)**8**(3) (1958), 236. - R C H Young, Review: L'Énumération Transfinie Vol. III. Etudes complémentaires sur l'ordination (1954) and Vol. IV. Notes sur les sujets controversés (1954), by Arnaud Denjoy, The Mathematical Gazette 39 (327) (1955), 72.
- A P Yushkevich, Letters of A Denjoy to N N Luzin (Russian),
*Istor.-Mat. Issled. No.***25**(1980), 362-368; 381.

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Written by J J O'Connor and E F Robertson

Last Update July 2020

Last Update July 2020