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.

28 Continuity of a function. Bk. I. ch. V.
call it h', is an interval for each point from a to 6, within which
the absolute difference between values of the function is certainly
smaller than 6. For, to an arbitrary point x occurring, ex. gr. in the
x1 x2 interval, belongs x + h' which is at most in the x2 x3 interval;
now we have
abs [f(xl) - f(x)] < i, abs [f(2) - f(x)] = -3
abs [f (x2) - f (x - +')] < -3 ~, therefore abs [f(x+h') - f(x)] <.
We have thus learned, that for each continuous function on e finite
value h can be assigned, which is sufficient for every x in the entire
interval from a to b in order that it may satisfy the inequality
f(x +& Oh) -f(x) < A,
d being given arbitrarily. In consequence of this property Heine (see
note p. 14) has called every continuous function uniformly continuous in its interval.
We can also state the result of these investigations thus: Any
continuous function accomplishes any assignable finite change in its
value only within an assignable finite interval; whereas with a discontinuous function a finite change takes place in an arbitrarily small
interval.
17. Points at which the criterion of continuity is not satisfied
are called points of discontinuity. Thus, for instance, the function
x —a _ i
y -.= 1- (e > 1)
ex- + 1
is discontinuous for x = a, as we find in calculating its value by
putting x = a + 7 or x = ao -, and making h converge to zero.
In the former case it is at once
1
e —
e + 1
For h = 0 this is a quotient whose numerator and denominator increase in a determinate manner beyond any assignable limit; yet it
approximates to a fixed finite value. For as it can always be put
equal to
1- ea A. -- -T
l-+ e h
we see that ultimately the values of the second factor differ arbitrarily
little om uity; because as decreases, e i always a decreasing
little from unity; because as h decreases, e -' is always a decreasing