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.

250 The definite integral as the limiting value of a sum. Bk. III. ch. VI.

c b c
ff(x) d X -J f(X)CIX + f (X) I X,
a a b
therefore:
b c c c b
(x) d x = (x)dx - x '(x) dx = (x) d x + (Jfx)dx.
a a b a c
IV. The sum of integrable functions is itself integrable, we have:
b b b b
fl (X) ~ f2 ()... f( }x=J fi (x) dx f2 (x) dx ~+ + (x) dx.
a a a a
V. The product of two or more integrable functions is itself
integrable.*)
We must remember that the foregoing discussions only deal with
functions that do not become infinite. An extension of theorem V.
will be found in ~ 149.
In the interval cd let the value of the greatest fluctuation of the
function q (x) be DL and of 4'(x) be Dp'. We have by hypothesis:
p-=n p-=n
cdDpDp,== 0 (dp for n ==o.
The product (x) is subject in the same interval to uctations,
The product p(x). O(x) is subject in the same interval to fluctuations,
which, if x + -Odc and x + O'dp denote the places of its greatest
and least values, are measured by the difference:
(p (x+ ocdp)(x + o0) - gp(x+ O'd) (x + ' dp)
= p(x + Od c) {4(x + QOcd) - f (x + O'dp)}
+ 4(x + O'dp){p(x + oGd)- (x+O'{ )}.
This form shows that the fluctuation of the product is certain not to
exceed GC GpD' + G'Dp, if Gp and Gp' denote the greatest absolute
amounts which the functions gp and 7 assume in the interval dp.
When G and G' are the greatest of all the absolute amounts which
these functions assume in the entire interval of integration, we have:
p=n
{p(x + Odp) '(x + p) - p (x + O'Cp) 4 (x + O'd) }dp
P=1
pp=n pn
< G dpDp' + G''pDp
therefore in consequence of our hypothesis, it vanishes. Q. E. D.

*) Du Bois-Reymond, Journal f. Math., Vol. 79, p. 21.