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.

170

The definite and the indefinite integral.

Bk. III. ch. I.

of one of them; we shall also find that the uniform continuity of the
quotient of differences arises from it.
The propositions we have stated can be proved by the following
considerations:
1. When a continuous function has throughout the whole of
an interval a positive progressive differential quotient, the values of
the function increase in this interval and its initial value is less than
its final value.
At each point, at which a continuous function has a progressive
differential quotient different from zero, a progressive interval A x
can be assigned, within which the difference f(x + O Ax)-f(x)
does not change sign, ~ 20. Hence, if the function were to decrease at
a point instead of increasing, f(x + O Ax) - f(x), and therefore also
the differential quotient, should be negative. Moreover the case is
inconceivable, that while x converges to a determinate point x' in the
interval, A x should fall below any assignable limit. For, let us form
the difference f(x' - ~ + Ax) - f(x' -,) and make E converge to
zero, then in case Ax were to converge to zero, this difference should
become zero in consequence of the continuity of f, but since at the
point x' there is a positive differential quotient, there must at any rate
be an assignable interval h, within which f(x' + h) - f(x') remains
positive. Therefore f(x' + h) - f(x' - e) also is positive, however small
E is chosen.
2. When a continuous function assumes the same value at the
extremities of an interval, throughout which it has a determinate
progressive differential quotient that is continuous in the entire interval,
there must be a point, at which the differential quotient vanishes.
Since the function attains the same value at the extreme points,
unless it remain throughout constant, it must undergo alternation in
its continuous increase or decrease, i.e. must have points at which
its differential quotient is positive and points at which it is negative.
But as this latter is continuous, there must be between these a point
at which it vanishes.
3. For every continuous function f whose progressive differential
quotient fi is also continuous, we have in an interval from x0 to X
the equation:
f(X -o f Xf + o (X - x0) (O < o < 1)
x -
For, if we denote the value of
f(X)- fx) by K,
X - Xo
then (p x) =- {f(x) Kx} - {f(xo) - Kx0o}
is a continuous function of x, which has the same value, zero, at both