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.

~ 79-82.

Continuity of an explicit function.

129

or of r and p - let us ask what is the analytical characteristic
that such a function is unrestrictedly continuous in a domain for which
it has determinate values. The foregoing discussions indicate that
it must be possible to find surrounding any point z at which w
is to be continuous, a connected domain of two dimensions of finite
extent however small (the Neighbourhood of the point z), to which corresponds a connected domain of w; i. e. the quantities u and v must vary
continuously when the quantities x and y, or r and 9 vary continuously;
in other words: u and v must be continuous functions of both the
real variables x and y, or r and 90 (~ 52). When the quantities u and
v are expressed as functions of r and 9g it must however further be
remarked, that should the function w be one-valued, an increase of q
by multiples of 2z must not alter the values of these functions.
Denoting the increment of z by Az = Ax + iAy and putting:
ow + A w = (u + Au ) + i(v + Av) = f(z + a ),
then in whatever way Ax and Ay converge to zero, we must have
Lim Au = O and Lim Av - 0.
These two conditions are combined in the single statement:
The function t =- f () is continuous at a point z, when this point
can be included within a domain such that the modulus of the difference:
mod [A w] == mod [f(s + A) - f(z)] =- V/A 2 + A V2
for every point -- A in this domain, shall be less than any arbitrarily
small prescribed number 6.
82. Turning now to the formation of definite examples of functions of a complex variable, before all things we restrict ourselves to
such as admit of the calculation of one or more values of w by
a given formula for each value of the argument z. The most general
instrument our previous investigations have provided for this purpose
is the series of powers, which embraces the explicit rational or
irrational functions.
Of such series we have already become acquainted in the domain
of real quantities with the exponential series, and its inverse, the
logarithm; to these we can reduce trigonometric and circular functions
(~ 67 and ~ 74).
Accordingly we propose to ourselves the task of studying in the
complex domain: first the explicit rational and irrational functions,
next the exponential function and its inverse the logarithm, and then
in general the properties of functions expressed by series of powers.
These problems form the basis of the general Theory of Functions; to
them the following investigations always return, inasmuch as in their
progress the methods for.the complete solution are gradually attained.
IHARiNACi, Calculus. 9