Complex Analysis/Complex Functions/Continuous Functions
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We now introduce the fundamental concepts of limits and the continuity of functions.
Let f(z) be a complex-valued function defined on a subset 
of the complex plane. We then say that the limit of f(z) as z tends to an accumulation point z0 of exists and equals the complex number L if, for any real number ε > 0 we can find a real number δ > 0 such that | f(z) − L | < ε for all 
that satisfy | z − z0 | < δ, and we write this limit as
.
An alternate but equivalent definition can be made using open sets: we say that the limit exists and equals the complex number L if, for any real number ε > 0 we can find a neighborhood O of z0 such that | f(z) − L | < ε holds for all 
. Since the first definition is easier to work with, we will often use that one.
A function w = f(z) is called continuous at z0 if f(z0) is defined and 
.
If a function is continuous at every point in a set, we say it is continuous throughout that set. Also, we will simply say that a function is continuous if it is continuous everywhere.
