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.

46

Differential quotient of a function.

Bk. 1. ch. V~

we have also Lim - 1. But on the other hand f' (x)=- 1 - sin (x)
+ 2 cos (x2) is for x- = o indeterminate.
e) If f(x) has a finite determinate value for x --, then for
every finite value of h we have Lim {f(x +- 7 f(x}) 0. This is also
X=z o0
to be regarded as value of the differential quotient at x =- c; it coincides with the value Lim {f'(x)} provided f'(x) changes continuously
to any determinate limiting value for x = oc as x increases arbitrarily.
f) The differential quotient can be indeterminate at all points of
a continuous function, if ex. gr. the difference f(x + Ax) - f(x) everywhere in an arbitrarily small interval Ax undergo change of sign
f (X - Ax) - f (x)
without the amount of the quotient of differences f(- ______
converging to zero; this is a case, in which the 'function cannot be
fixed under the figure of a curve, on the basis of the formula in
accordance with which it is to be calculated, because the interpolation
of more and more points ultimately displaces the angles of the polygon
immeasurably, whereas it displaces the sides measurably.
g) In representing to ourselves how motion goes on in nature we
presuppose no discontinuities, neither in regard to the places which the
moving body occupies, nor in regard to the direction and magnitude
of its motion. In some phenomena however (under the influence of
blows) the changes are effected so rapidly, that we look upon the process as a discontinuous one.
The Continuity of a function was first precisely defined by
Cauchy (1789-1857) (Analyse Algebr. 1821), to him we owe the
foundation of the differential calculus generally in the form in which
it is here developed. Riemann (1826-1866) directed attention to
continuous functions which have within an arbitrarily small interval
infinitely many points at which the progressive and the regressive
differential quotients are indeterminate. Weierstrass was the first
who gave an example of a continuous function having in no point
a determinate value of the differential quotient either progressive or
regressive (communicated in a paper of Du Bois-Reymond Journ.
f. Math. Vol. 79). Here the function appears as limit of a series of
functions, whose values ultimately differ arbitrarily little, while the
same is not true for the values of the differential quotients which on
the contrary vary between arbitrarily great positive and arbitrarily
great negative values.