Un texte d’Euler sur les fonctions continues et les fonctions discontinues, véritable programme d’organisation de l’analyse au 18e siècle. (A text of Euler concerning continuous and discontinuous functions; the real program of the organisation of analysis in the 18th century). (French) Zbl 0628.01010

Cah. Sémin. Hist. Math. 9, 23-97 (1988).
This paper gives a French translation of a memoir by Euler concerning the nature of functions and the scope of analysis. The author includes a detailed and illuminating discussion of the ideas of the memoir. Two important problems in the analysis of the mid-eighteenth century were the validity of the functional method; and the distinction between continuous and discontinuous functions (these terms had then different meanings from the present ones). Underlying both these problems was the still unclear definition of function. The central idea of Euler’s memoir is that analysis must be concerned with all the functions that arise as the general solutions of differential equations; in the case of partial differential equations general solutions of necessity involve arbitrary functions, including those (discontinuous in the 18th century sense) which are piecewise regular. In this way the controversies regarding the arbitrary functions which appeared in the solutions of problems such as the vibrating string might be settled.
Reviewer: G.Smith


01A50 History of mathematics in the 18th century
26-03 History of real functions

Biographic References:

Euler, L.
Full Text: Numdam EuDML