An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

244 The definite integral as the limiting value of a sum. Bk. III. ch. VI.
finite number of separate points (~ 105). For, these separate points,
suppose n in number, can be included within arbitrarily small intervals 6, such that when D denotes the largest value among the sudden
changes, these intervals of discontinuity do not contribute more to the
sum S than n6D. Since n and D are finite, d can be chosen so as
to make this product arbitrarily small.
We can likewise see that, at each of an arbitrary finite number
of points, the function can have, within an interval however small,
infinitely many maxima and minima with finite fluctuations, (as ex. gr.
sin 1 at the point x= a), or even that it can be left altogether
x - a
indeterminate, i. e. that we are at liberty to attribute any finite value
whatever to the function at such a number of points, without the
value of the definite integral being thereby altered.
Third: when at an infinite number of points the function is
discontinuous or indeterminate between finite limits, or else; within
an interval however small, has infinitely many maxima and minima
with arbitrary finite fluctuations; provided this infinite system of points
answer to a certain definite description. Into this we shall enter in
the next Paragraph because the investigation presents an occasion for
us to extend considerably our conception of a function.
144. To grasp the conception of an infinite number of points,
we must first of all dwell upon the difference: a finite length contains infinitely many points, but infinitely many points do not necessarily fill up a length, or in purely arithmetical language: the continuous series of numbers between any two limits, contains infinitely
many numbers between these limits, but yet infinitely many numbers
between two limits do not fill up the series of numbers. In order to
characterise this difference we introduce the following definitions:*)
Naming the interval from x - to x + -, whose length is any
arbitrarily small finite quantity 2e, the neiohbourhood of a point x,
we shall call an infinite multiplicity of points a discrete set or mass
of points, when it is possible to include all of these points within neighbourhoods whose sum can be made smaller than an arbitrarily small
length, while the number of the neighbourhoods caiincrease arbitrarily..) The investigation of infinite sets of points first given concisely (1871) by
G. Cantor, Math. Annal., Vol. V, is developed also in Dini: Fondamenti per la
teorica delle funzioni di variabili reali. Pisa 1878. The above distinction of
discrete and linear sets of points differs however fiom Cantor's definition of sets
of the first and second species (Math. Annal., Vol. XV, p. 2). In strictness, by
the phrase "discrete set" of points or values, we imply that for the problems
of the integral calculus such a set has the same property as a finite number of
separate points or values, often using it briefly whether the number is finite or
infinite. See Ex. 1).