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.

248 The definite integral as the limiting value of a sum. Bk. ITI. ch. VI.
the integral of the function so defined is zero. The sets of points in
question we may ex. gr. conceive formed by the series:
+ (41)In, + (a))m+1, + (1)+2 ' *, etc..
For, the points at which the discontinuities exceed some given finite
number, form always only a finite number of discrete sets of points.
The sum of their neighbourhoods becomes arbitrarily small.
3) The first example of a discretely discontinuous integrable
function was given by Riemann.*) Let (x) denote the positive or
negative excess of x over the nearest integer less or greater; when x
is midway between two integers, let (x) = 0. The series:
f() i f- 22 + 32 +m2
f(x) = +L (2)- (3..) m
converges; for, -.+ - -+ t- +' is convergent (~ 47) and its sum
found from the expansions of tan x and cot x is I-,2; also the value -I
is a superior limit of (tmx). Each term of the series f(x) is generally
continuous, only when 2mnx = an odd integer p, neighbouring values
in the function (mnx) differ from each other qluam proxime by 1.
When x -Pn this takes place not only with the term (mnx) but
also with the terms (3mx), (5m x), etc.. Hence follows: When x is
of the form --- where p is prime to mn:
2m 2 7m2
f(x + O)- f(x) 6m2 f(x - 0) =-f(x) + 16 - 2 -For, when x = - the terms named contribute nothing, they vanish;
while when x begins to increase, they increase each quam?proxime
by - -~, and when x decreases, each increases by -+ -. But::1 m2 { 1 2 522 1 321 2 3
1 72 5 2 2A 12
= 2
22 \n 6 24f 16m
For each rational value of x, that in its lowest terms is a fraction
with an even denominator 2m, there is therefore a discontinuity
of f(x); and thus, infinitely many between any two limits however close.
But the number of such discontinuities whose value exceeds a given
limit is finite; for, if 8 must be >, n m must be <-. But in
8rn'~~~~~~~~~~~~2 /,

*) Gesammelte Werke, p. 228.