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.

172

The definite and the indefinite integral.

Bk, IIl. ch. I.

fi(x + ) - f, (x) < 6
by any assignable value of Ax, but this would be contrary to the
continuity of f1.
101. The fundamental problem of the Integral Calculus consists
in the inversion of the problem of differentiation; it may be expressed:
Any arbitrary unique function f(x) being given in the interval from
x = a to x = b; it is required to find a continuous function F(x)
possessing the property that its derived function is identical with f(x)
for all values from x = a to x = b.
Regarding the function f(x) we make here the following restrictive
hypotheses: first, f(x) is to be throughout the entire interval finite;
second, f(x) is to be throughout the entire interval continuous, or if
not, its discontinuities must be finite, and, however numerous, they
must occur only at isolated points.
When f(x) is a continuous function, the required function F(x),
if it exist, is such that its progressive and regressive differential
quotients coincide everywhere in the interval. But when f(x) is
discontinuous at separate points, so that at any such point the values
Lim f(x + 6) and Lim f(x - 6), that by hypothesis are determinate,
are different for 6 = 0, the function F(x) must be such that its
progressive differential quotient at this point is equal to f(x - 0), and
its regressive is equal to f(x - 0); these abridged notations being
employed for the limiting values above named.
Now the first question to be answered is whether under these
conditions and with these data the problem is definite or not; that
is, whether there are not different continuous functions whose
derived functions coincide in the interval from a to b. Suppose that
besides F(x) a second function ) (x) were found whose differential
quotients in the interval a to b likewise equal f(x); then <D(x) --.F'(x)
is a continuous function having its progressive and regressive differential
quotients throughout that entire interval zero. Such a function can
only be a constant, as was proved in the last Section. Hence:
0(x) = F(s) + Const.,
i. e. all continuous fmnctions, that have the same determinate values of
the progressive and regressive differential quotients respectively in an
interval, differ from each other only by an additive constant whose
value is arbitrary.
This result can also be stated as follows: There is only a single
continuous function whose differential quotients coincide withf(x) in the
interval a to b, and which has a determinate, arbitrarily chosen value
at the point x= a. For, the additive constant is uniquely fixed by
establishing a value of the function at the point x =a.