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.

~ 144. 145.

Discretely and linearly discontinuous functions.

247

By "fluctuation" of the function is meant the magnitude of the
breaches of continuity, or, the difference of the limits between which
the indeterminate values lie, or lastly the difference between the
maximum and minimum values.
On the other hand we call a function linearly discontinuous,
in which such points form a linear set of points. Now it is easy to
see from Riemann's Theorem that within its interval:
A discretely discontinuous function is integrable.
For, a being a prescribed arbitrarily small finite number, we can
carry the partition of the length ab so far, that in the partial intervals
generally the fluctuations become less than a, while s, the sum of the
neighbourhoods of all the points at which the fluctuations exceed a,
can be arbitrarily diminished. Let m be the greatest value among
these fluctuations, then:
y=-n
d dD < a(b - a) + sm.
t =1
But the sum on the right can be diminished arbitrarily, since a may
be assumed arbitrarily small, and likewise s in consequence of the
property of a discrete set of points.
Linearly discontinuous functions are not integrable.
Examples of functions infinitely often discontinuous that are
integrable.
1) Let the value of the function f(x) be zero everywhere in the
interval from 0 to 1, except in the infinite series of points:
i, ()2, ()3j (;)4,... (1)n
in which its value is to be -. This function is infinitely often
discontinuous within an arbitrarily small interval from zero; but the
sum of the intervals in which the fluctuations are - can be made
arbitrarily small. The value of the integral is determinate, it vanishes.
2) We can likewise construct a function that is integrable, although
it is discontinuous in every interval however small, and though the
number of points, at which it has discontinuities greater than some
finite number, is not finite. Determining, ex. gr., that the function
f(x) is to vanish generally in the interval from zero to unity, but
yet that at all points of a discrete set of which the point ~ is the
derived set its value is to be ~; at the set of points whose derived
set are the points a, 2, its value is to be a; at the set of points
with the derived set -,,,,, its value is to be {-; and generally:
if p be a prime number and q denote each number smaller than p, at
the set of points with the derived set, its value is to be
p~ 2o