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.

~ 19-20.

Theorems concerning continuity.

33

value and whether it is a continuous function at this point also, i. e.
whether it arrives at the same value by proceeding from x > a as from
x < a, must be decided by particular methods. The most important
case of this kind will be investigated in the following paragraphs: its
solution gives rise to a general method.
d) If u =- p(x) be a continuous function of x, and y =- f(u) a continuous function of u, then y is also a continuous function of x.
For in order that
abs [f (9 (x + h)) - f (p (x))] may be < 6,
the change of x must be so determined, that
abs [(p (x + h) - p(x)] < e,
E denoting the quantity for which the condition
abs [f(u + E) - f(u)] < 6
is fulfilled for the continuous function f. But by hypothesis such a
value h can be assigned.
20. Having ascertained the remarkable points in the course of a
function, we proceed to investigate a measure for the way in which the
function changes its values when the independent variable is increased
or diminished. The leading idea in the Differential Calculus is
the establishment of this measure with full mathematical precision.
Consider two values of the function, belonging to two different values
of the argument x and x + Ax, Ax denoting here the increment of x
must not be mistaken for a product and can be chosen > or < 0; let
the corresponding values of the function be denoted by y and y + Ay,
then A y can be calculated from the equation:
(I) Ay =f(x +A x)-f(x).
The difference on the right assigns the magnitude of the change of y
when the independent variable alters from the value x to the value
x + Ax.
Now this change of y has to be compared with that of x. Keeping
the value of Ax unchanged, the intensity of the change in the value
of the function becomes greater when the value of Ay is increased.
On the other hand, for a given value of Ay the intensity of change
would be increased if Ay were produced by a smaller Ax. Thus the
quotient of differences
(II) Ay _ X f x + Ax)-; (AX > )
w(II) ^^Ax x( <o)
gives a measure for the average (mean) intensity of the change in the
interval from x to x + Ax. If the function changed uniformly in the
interval, that is, if equal values of Ay always belonged to equal
values of Ax, then if the intensity, or rate, of change were that given
in equation (II), through the entire interval Ax it would give rise to
HARNACK, Calculus. 3