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

12.4. Continuity 233
it followed that
rm <
and the inequality had been obtained.
12.4 Continuity
m−1
∑ δ (εk − εk+1) + εmδ = δε0,
k=0
Just as had been the case with CAUCHY’S proof of the binomial theorem, the concept
of continuity played an important role in ABEL’S proof. In his paper on the binomial
theorem, ABEL gave rudiments of a different rendering of the theory of interaction
between the concepts of continuity and convergence. ABEL’S definition of continuity
seems to closely resemble CAUCHY’S (see page 216), although it may be noticed that
ABEL’S definition is only formulated in the terminology of limits.
“Definition. A function f (x) shall be called a continuous function of x between
the boundaries x = 0 and x = b when for any arbitrary value of x between these
limits, the quantity f (x − β) for ever decreasing values of β approach the limit
f (x).” 27
Although their definitions of continuity are almost identical, ABEL and CAUCHY
attributed slightly different meaning to their concepts when they were employed. The
apparent ambiguity concerning the order in which quantification is to be made in
CAUCHY’S definition was resolved in ABEL’S persistent insistence on point-wise def-
initions. ABEL’S definition as stated seems just as susceptible to the ambiguity as
CAUCHY’S, but ABEL throughout interpreted it to mean that a function is continuous
at a point x ∈ [0, b] if f (x − β) → f (x) as β → 0.
Combining continuity, convergence, and power series. The fourth and fifth theo-
rems of ABEL’S binomial paper provided two important combinations of the concepts
of continuity and convergence. The fourth theorem, Lehrsatz IV, stated and proved the
continuity of a power series in the interior of its interval of convergence, while Lehrsatz
V attempted to provide a rigorous replacement for what Cauchy’s Theorem (see page
217) had promised but not rigorously delivered. At this point, Lehrsatz IV together
with its proof will be described first, and Lehrsatz V will be postponed to be discussed
in its proper context of ABEL’S famous Ausnahme or counter example to CAUCHY’S
theorem (see section 12.6). At that point, the strong internal relations between the two
theorems will also be described and explained.
As his fourth theorem, ABEL stated and proved the following result which has
become a classic of the theory of series and is often associated with ABEL’S name.
27 “Erklärung. Eine Function f (x) soll stetige Function von x, zwischen den Grenzen x = 0, x = b
heißen, wenn für einen beliebigen Werth von x, zwischen diesen Grenzen, die Größe f (x − β) sich
für stets abnehmende Werthe von β, der Grenze f (x) nähert.” (ibid., 314).