After being given leave of absence from school teaching, Pompeiu went to Paris in 1898 to continue his mathematical studies. After obtaining his licence he continued to undertake research for his doctorate with Henri Poincaré as his thesis advisor. He submitted his thesis Sur la continuite des fonctions de variables complexes Ⓣ and defended it on 31 March 1905 before a committee chaired by Henri Poincaré which included Émile Picard, Gabriel Koenigs, Paul Appell and Édouard Goursat. The thesis was published in Paris in 1905 and, in the same year, was also published in the Annales de la faculté des sciences de Toulouse. This thesis was an important work and we shall discuss something of its content and importance below. For the moment let us continue to outline Pompeiu's career.
He returned to Romania in the autumn of 1905 where he was appointed as a lecturer in mathematics at the University of Iasi. He was promoted to professor of mechanics at Iasi in 1907 and remained there until 1912 when he moved to the University of Bucharest. His appointment in Bucharest was an important one for he became the successor of Spiru Haret who was one of the two most famous of the first generation of Romanian mathematicians. The second of the pair of famous first generation of Romanian mathematicians was David Emmanuel and, when Emmanuel retired in 1930, Pompeiu was named Professor of the Theory of Functions to succeed him. A small selection of the papers published by Pompeiu following his doctoral thesis are: Sur les fonctions dérivées Ⓣ (1907), Sur un Exemple de Fonction Analytique Partout Continue Ⓣ (1910), Sur une équation intégrale Ⓣ (1913), Sur les équations fonctionnelles des polynômes à variables réelles Ⓣ (1934), Du point à l'infini comme point singulier isolé Ⓣ (1938), Remarques sur l'équation de Riccati Ⓣ (1940), La géométrie et les imaginaires: démonstration de quelques théorèmes élémentaires Ⓣ (1940), and De la définition du pôle en théorie des fonctions Ⓣ (1940).
We promised to return to discuss Pompeiu's doctoral thesis. The motivation for the research he carried out was a question about the singularities of uniform analytic functions posed by Painlevé in Leçons sur la théorie analytique des équations differentielles Ⓣ in 1897. The difficulty arose when, also in 1905, Ludovic Zoritti wrote a doctoral thesis in which he claimed to have proved that a uniform analytic function cannot be continuously extended on the set of its singularities. However, Pompeiu's doctoral thesis written in the same year proved the existence of certain analytic functions which could be extended continuously on their set of singularities even though this set had positive measure. Clearly both results could not be correct and the difficulty was resolved in 1909 when Denjoy confirmed that Pompeiu's results were correct, and he found the error in Zoritti's theorems. Pompeiu's examples had been constructed using ideas due to Koepcke and they were difficult to understand. However, in 1907 Pompeiu had clarified the whole situation by constructing simpler examples in his paper Sur les fonctions dérivées Ⓣ which we mentioned above. The functions which he constructed in this paper are now called 'Pompeiu functions'. Marcus discusses these functions in  and traces their influence on the development of analysis.
There was another important idea in Pompeiu's doctoral thesis, namely the distance between two sets which he called the 'écart' and 'écart mutuel' which :-
.... allows Pompeiu to see the compact subsets in the plane as the elements of another set and to define in a natural way limits, closure, etc. for this "set of sets". Consequently, Pompeiu is also considered as one of the founders of the theory of hyperspaces.McAllister, in , makes a detailed historical study and agrees that Pompeiu must be considered as one of the founders of the theory of hyperspaces. The distance between sets is now usually called the 'Hausdorff distance' since the idea appears in Hausdorff's famous book Grundzuege der Mengenlehre Ⓣ (1914). In this work Hausdorff gives a slightly different definition of distance between sets but he also credits Pompeiu's work and shows that the two definitions give the same topology. McAllister writes on page 310 and 311 of :-
... [Pompeiu] may with some justice be said to have invented hyperspaces, and Hausdorff's use of them in 1914 in his treatise Grundzuege der Mengenlehre has made them very well known ....I have found no evidence of the Hausdorff metric itself before Pompeiu's thesis.
Petru Mocanu has written of Pompeiu's contributions in . He writes:-
There is no doubt that Pompeiu's preferred area was analysis, especially complex analysis, but he achieved remarkable results in other areas such as mechanics. Pompeiu initiated the theory of polygenus functions as a natural extension of analytic functions. He introduced the notion of a special type of derivative, the areolar derivative of a complex function, extending the Cauchy formula which today is sometimes called the Cauchy-Pompeiu formula. In a short paper in 1929 [Sur certains systemes d'équations linéires et sur une propriété intégrale des fonctions de plusieurs variables Ⓣ] he proved that if the double integral of a continuous function takes the same value over any square of given side, then the function is constant. This simple remark has led to many interesting problems in analysis known as the problem of Pompeiu.The authors of  state that around 1000 papers have been written which cite this 1929 paper by Pompeiu - a remarkable achievement. Among other topics on which Pompeiu published research articles we mention interpolation theory. On this topic he wrote six papers spread throughout his career - they are discussed in .
In  Iacob discusses Pompeiu's design for a three-year mechanics course in Bucharest, using both students' notes and university records. After World War I ended, Pompeiu organised the Mathematics Seminar at the University of Cluj. He was the first director of the seminar which he modelled on the seminar at the Collège de France. Finally let us mention that Pompeiu, along with Petru Sergescu, founded the journal Mathematica (Cluj) and he was its first editor.
Pompeiu was honoured with election to the Romanian Academy of Sciences in 1934.
Article by: J J O'Connor and E F Robertson