RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

216 Chapter 11. CAUCHY’s new foundation for analysis
and if the terms um and v k were positive, the difference (11.6) would converge to zero
as a consequence of the theorem for series of positive terms. If the terms were not all
positive, CAUCHY could still apply the theorem to the corresponding series of numer-
ical values ρn = |un| and ρ ′ n = |vn|. Therefore, in this case, the difference (11.6) would
still converge to zero and yet majorize the series of the original terms. Thus, CAUCHY
had established the convergence and the product equation for absolutely convergent
series of real terms.
CAUCHY’S concept of continuous functions. CAUCHY’S concept of continuous func-
tions (see below) was among the main innovations of his new calculus. There, in the
definition and in the proofs, his new foundation on an algebraic concept of limits
played its most important role. In the eighteenth century, EULER had used the term
continuous to indicate that the function was defined by the same analytic expression
throughout its domain. 21 However, in the Cours d’analyse, CAUCHY took it upon him-
self to completely redefine the concept of continuous functions in order to capture a
different property of the functions: their continuous, unbroken variation.
“Let f (x) be a function of the variable x and suppose that for every value of
x between two given boundaries this function always takes a unique and finite
value. If, starting from a value of x contained between these boundaries, one attributes
to x an infinitely small increment α, the function will receive the increment
f (x + α) − f (x)
which depends simultaneously on the new variable α and the value of x. Between
the two boundaries assigned to the variable x, the function f (x) will be a continuous
function of this variable if for every value of x between these boundaries, the
numerical value of the difference
f (x + α) − f (x)
decreases indefinitely with that [numerical value] of α. In other words, the function
f (x) remains continuous with respect to x between the given boundaries if, between
these boundaries, an infinitely small increment of the variable produces an infinitely small
increment of the function.” 22
21 (L. Euler, 1748). See (Lützen, 1978) and (Youschkevitch, 1976).
22 “Soit f (x) une fonction de la variable x, et supposons que, pour chaque valeur de x intermédiaire
entre deux limites données, cette fonction admette constamment une valeur unique et finie. Si, en
partant d’une valeur de x comprise entre ces limites, on attribue à la variable x un accroissement
infiniment petit α, la fonction elle-même recevra pour accroissement la différence
f (x + α) − f (x) ,
qui dépendra en même temps de la nouvelle variable α et da la valeur de x. Cela posé, la fonction
f (x) sera, entre les deux limites assignées à la variable x, fonction continue de cette variable, si,
pour chaque valeur de x intermédiaire entre ces limites, la valeur numérique de la différence
f (x + α) − f (x)
décroit indéfiniment avec celle de α. En d’autres termes, la fonction f (x) restera continue par rapport
à x entre les limites données, si, entre ces limites, un accroissement infiniment petit de la va-