Continuity

A function , defined over a domain , is said to be continuous in if for every point of we have The limit of a complex function is defined here as in the real case. If for any there is a such that we say that approaches the limit as tends to and write Thus, the definition of continuity amounts to the requirement that given any preassigned there is a such that for all for which . Geometrically this means that no matter what circle we draw with as center, it is always possible to find a neighborhood of which maps completely inside the given circle.

The notion of limit for a function is only a little more general than that of continuity. A function which possesses a limit at a point can be made continuous simply by altering its value at the point to coincide with the limiting value.

The definition of continuity may be given in other ways. A function is continuous at the point in its domain of definition if for every sequence in with we have Again, is continuous in if the real and imaginary parts, and , separately are continuous functions of and in . The reader may demonstrate the equivalence of these three definitions for himself.

If two functions and are both continuous in then evidently and are also continuous in . Furthermore, the quotient is continuous in any subdomain of in which .

The proof is a direct application of (2.01) . It is now easy to find large classes of continuous functions. From the fact that and are both continuous it follows that any polynomial is continuous in the entire -plane and moreover, that any rational function is continuous in any domain in which the denominator is not zero.

For a continuous function, the function values remain within an -neighborhood of for all values in a sufficiently small -neighborhood of where depends on both and . In general, it will not be possible to pick completely independently of the point . The function , for example, is continuous in the domain . But, given any , there is no fixed value of which can be employed for the entire domain; for, clearly, as approaches the origin we must let . In contrast, we say that a function is uniformly continuous if for every positive there is a such that for all points , in satisfying For a finite region continuity implies uniform continuity. In fact we may state more generally:

Setting we see that this result is a corollary of the theorem for real functions. ¹ 

There are numerous powerful techniques for representing a function by a convergent series of functions. It is therefore essential to have a method of determining whether a function given as the sum of a convergent series is continuous. Such a criterion is provided by the theorem:

As a corollary of this theorem and the first theorem from Section on Power Series we observe that a power series represents a continuous function in the interior of its convergence .