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.

~ 20.

The quotienzt of differences.

35

But if the limiting value is to be finite or determinately infinite,
then since when + 1 > 0 > 0 the denominator of the quotient is
only positive, the numerator must also always have one and the same
sign for the same sign of 0, that is,
f(x + Oa x) is either only > or only < f(x).
The function only increases or only decreases in comparison with its
value at x, while x increases; it no longer oscillates about this value
in an arbitrarily small interval, This is a necessary condition. But
the condition necessary and sufficient in order that the series of quotients of differences may have a finite limiting value, by ~ 6 is that
abs [ +Ax) - f(x) f (x+oAx) _ f (x) <, (1 > 0 ~ O)
Ax OAx
where 6 signifies an arbitrary number.
This inequality can be interpreted as follows:
Galling the numerical difference between the quotients of differences
belonging to the values Ax and OAx which depends on 0, a fluctuation of this quotient in the interval Ax, then this inequality asserts:
The necessary and sufficient condition for the existence of a determinate
finite limiting value consists in this, that for each number 6 however
small, a finite interval Ax can be ascertained, in which all fluctuations
of the quotient of differences are less than '.
But this amounts to saying, that in the neighbourhood
of any point at which the quotient of differences of the
continuous function has a determinate finite limiting value,
the quotient of differences is not only a continuous function
of Ax for every finite value of Ax, but retains this property also for Ax = 0.
If the quotient of differences has the property of only increasing
or of only decreasing in the interval Ax, while 0 converges to zero,
in other words: if at any point an interval Ax can be found in which
the quotient of differences has neither maxima nor minima, then as in the
Theorem in ~ 6 (cf. ~ 9), it has a limiting value either finite or determinately infinite. The possibility of no determinate limiting value arises
accordingly only in case the quotient of differences at a point assumes
in however small an interval infinitely many maxima and minima,
whose differences cannot be arbitrarily diminished. In this case we no
longer say that the quotient of differences is continuous inclusive of the
value A x = 0.
We can illustrate numerator and denominator of the above inequality
geometrically as follows:
Let us represent the function y= f(x) under the figure of a polygon
with any number of angles, interpreting the values of x and y as lengths
in a rectangular system of Cartesian coordinates. Then we perceive:
3 *