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.

~_15-16.

Geometrical representation of a function.

25

Our first enquiry is: what properties does the course of the dependent variable y exhibit while the independent variable x assumes all
possible values? There are three things of immediate importance in
the examination of any proposed function.
First: Is its course everywhere continuous or not?
Second: What singularities occur among the values which the
function assumes?
Third: What values does it take, when the independent variable
becomes infinitely great?
16. Let the explicit function y = f(x) be defined as a one-valued
function for a determinate interval from x = a to x -b, i. e. let one
and only one determinate value of the function without any exception
belong to each value of x. We call the function continuous on
both sides of any point x in this interval, when there are no sudden
changes in its values as we move from x to either side, that is,
as we form the values of the function belonging to values arbitrarily little greater or less than x. In a form applicable to calculation we state what is required thus: It must be possible for this
value of x, to find a finite number h, which only converges to zero
with 6 = 0, such that the absolute amount of the difference f (x -1- Oh)
- f (x) of the values of the function is less than an arbitrarily small
assigned number 8, 0 denoting a variable between the limits 1 and 0.
For, there is thus fixed about this point x, a region + h in which the
values of the function differ by less than the arbitrarily small quantity
6 from the value at x, and this excludes a sudden change of the
function at that point. The same condition can be stated in other
words: The value of the function for a determinate x must come as the
limiting value both from f (x + h) and likewise from f (x - h), when h
becomes infinitely small, i.e. converges continuously to zero, provided
this limiting value is completely determinate*); for if so, by the fundamental property of a series defining a limiting value there will be a
value of h for which abs [f(x _ h) - f(x)] < 6, and this absolute
amount will remain smaller than 6 as h converges to zero.
Thus for instance
sin (x ~+ h) - sin x = 2 sin (+ h) cos (x + h).
Here h can be determined so as to make the first factor on the right
side arbitrarily small, for the sine is smaller than its argument, the
second is finite for every x; accordingly by a suitable choice of h the
product can always be made less than an assigned number 4, whence it
results that the sine is always a continuous function (see p. 30).
*) Lim (cos X- ') = Lim (cos _-h) yet this function is not continuous
for x 0O.