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.

38

Differential quotient of a function.

Bk. I. ch. V.

the outset, that the existence of determinate finite values for the progressive and regressive differential quotients, involves in itself always
the condition of the continuity of the function at this point.
21. According as the progressive differential quotient is positive
or negative, the function increases or decreases for increasing values
of x at this point; and according as the regressive differential quotient
is positive or negative, the function decreases or increases when the
values of x decrease. But the distinction of the progressive and the
regressive differential quotient for a continuous function is unnecessary
in most of the cases we shall have to consider, for we have the theorem:
If in the neighbourhood on both sides of a point at which f (x) is
continuous, an interval A x can be found for every value of x, such
that the differences of the quotients of differences of this intervals fbrmed for all values between 0 and A x, remain in absolute amount
smaller than an arbitrarily small number 6, then the progressive diffcrential quotient is a continuous function of x and the value of the regressive one is identical with it.
Taking an arbitrarily small but finite quantity A, lay off the interval +h on the two sides of a point x; this represents the neighbourhood on the two sides. The condition then asserts, that the
difference
abs [ f(x+h+-Ax) -f( f(xx-h) f(x+7-+OAx) - f(x+-7 h)
Ax OAx
remains smaller than 6, while 0 and X assume all values from 0 to 1.
This condition can be put into the words: The quotient of differences is a continuous function of both variables h and
Ax; or it is a uniformly continuous function of h and Ax.
(For the reason of this phrase, see Chap. IX.)
Let us denote the progressive differential quotient of f(x) at the
point x by fi(x), then, by hypothesis, Ax can always be chosen so
small that
(1) (x ~ h) (x + h + A x-f f(x + ~h) (< 6).
In like manner in consequence of the hypothesis of our theorem we
have for the differential quotient at the point x
(2);(X) + + Ax) _ f( + _h) + (<
and it is of importance that in the entire interval the same value
of Ax is sufficient for a given value of 6. Accordingly the amount
of the difference f, (x + h) - f1 (x) is also less than 26, i. e. the progressive differential quotient is continuous in the neighbourhood of
the point x.
To prove the second part of the statement, let us denote the re