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.

Third Book.

Integrals of functions of real variables.
First Chapter.
The definite and the indefinite integral.
100. Before considering the fundamental problem of the Integral
Calculus we must make ourselves acquainted with a theorem which
is supplementary to the propositions proved in ~ 21 and ~ 22.
From the Theorem of the Mean Value in ~ 22 we deduced that:
A function whose progressive and regressive differential quotients
vanish everywhere in an interval, is continuous in this interval and
in fact is constant. The example mentioned in ~ 17 shows that this
proposition does not admit of the enunciation: If the progressive
differential quotient vanish at each individual point in an interval, the
function is constant. For, the discontinuous function y = G(x), where
G signifies the greatest integer number contained in x, is discontinuous
at the points 1, 2, 3...; and yet we must admit that its progressive
differential quotient is zero at each individual point. For, however
near, ex. gr. we may assume x = 1 - to be to the point 1, still
an interval Ax < E can be assigned such that x -i- Ax < 1, therefore
G(x - Ax) -- G(x) _ 0
0.
By the help of the Theorem of the Mean Value, however, we can
perceive that a con t i nuous function, whose progressive differential
quotient vanishes everywhere in an interval, is constant, and at the
same time prove the Theorem of ~ 21 in the form:
When in an interval in which f(x) is continuous, its progressive
differential quotient is also a continuous finction of x, there exists
everywhere in this interval a determinate value of its regressive
differential quotient that is identical with the progressive.
Whereas therefore the uniform continuity of the quotient of
differences in regard to x and Ax formed previously the hypothesis
whereon rested both the identity of the two differential quotients and
their continuity, here that identity is to be deduced from the continuity