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.

~ 16. 17.

Uniform continuity. Discontinuities.

29

fraction. Determined thus, the value of the function for x = a is a.
But if we put x =- a- h, then
1
e -
y=a a
e h +1
hence for h- = 0 the limiting value is - a. Thus the function proceeds up to the value - a, when x beginning from a lower value
increases to a, but there the function suddenly leaps to the value + a
for the same x =, henceforth it decreases continuously as x increases; accordingly, for x ==a the function ceases to be one-valued.
The function already mentioned in ~ 13: y = G (x) where by
G(x) is meant the greatest integer contained in x, is when investigated
for positive values up from x == 0, a function everywhere one-valued,
which is discontinuous at the points x 1, 2, 3,..., the values
of the function suddenly changing from 0 to 1, from 1 to 2, and so
on. At all other points, however near they lie to the points of discontinuity, the function is continuous; we can even say, that for each
distinct value of x, we can choose h so as to make G (x + Oh)
- G (x) < ', only the value of h falls below any assignable limit
when x comes arbitrarily near a point of discontinuity: the continuity
ceases to be uniform; but we cannot make G (x - Oh) - G(x) < 6
when x is one of the points of discontinuity, whence we see that the
first inequality alone is not a sufficient condition of continuity.
But further, those points are also styled points of discontinuity, at
which the value of the function itself exceeds any assignable limit,
or at which it is quite indeterminate, because at such points also the
condition of continuity as above formulated cannot be satisfied. For
any given function it is then important to investigate, how it behaves
in the neighbourhood of such a point.
Thus, for instance, Y = -1 will have a negative value for
x < a, whose amount becomes greater the more x approaches the
value a; as soon as x has become a little greater than a, the amount
becomes positive and arbitrarily great; here therefore there is a change
from - oc to + oo, or
f (a + h)= and f (a - )-
present as h decreases, numerical series which become determinately
(~ 9) positively and negatively infinite. The same thing can be seen,
from the geometrical definition, to be the case with tan x when x
= ~z. On the other hand, the function y = ( )2 is at both sides
of a positively arbitrarily great. Continuity however is maintained for