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.

~ 142. 143. The general problem of integration and its conditions. 243
our process so far, that the value Sn of the sum belonging to the
former differs from its limiting value S only by the arbitrarily small
quantity E, whereas Sm' likewise differs from its limiting value S' only
by 6'; for, as was proved at first, every partition must lead to a
definite limiting value provided there be one for any such partition.
If now we imagine the two partitions combined into a single one
and form the corresponding sum Sm,n, relative to this single partition
resulting from their combination, this may be regarded as a step forward
as well in the series Sn as also in the series Sm'; hence Sm,, only differs
from S by a quantity X < E, and from S' by a quantity A' < E':
S =Smn,n +? S = Sn ~ ^\
The absolute difference S- S' will therefore not be more than V1 + r',
it is less than the arbitrarily small quantity E + -'; the limiting values
S and S' consequently are identical. The second question also is
answered by the same process. If we consider a succession of different
independent partitions: in the first let each of the intervals be less
than 6(1), and the sum of their products by the fluctuations be less
than e(1); in the vth let each of the intervals be less than d(), and the
sum of their products by the fluctuations be less than,(v); further let:
qS(1), S(2)...S(.
be the respective values of the sum; we can again combine the vth
partition with the first and regard this combination as a continuation
of the first as well as of the vth partition. Denoting the value of
the sum relative to the combination by S', we have:
S' == S + (< (1)), S' = S(Y) + (< (1")).
Therefore S() and S(v) differ by less than the arbitrarily small quantity
(') +- E(); i. e. the series of the sums S has a determinate limiting value.
The limiting value of the sums S is called the definite integral and
is denoted by the symbol:
b
ff(x) dx.
a
143. The condition thus established is fulfilled:
First: when "the function f(x) is throughout; continuous; this
was proved directly in ~~ 102 and 103. For, in this case a quantity
6 can be found such that at all points, in intervals that are equal to
or less than 6, the fluctuations of the function:
abs [(x) - f(x + 06)]
are smaller than an arbitrarily small number a. Hence we have:
dD < (b - a)a.
Second: when the function f(x) has finite discontinuities at a
16*