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 of exists and equals the complex number L if, for any real number we can find a real number such that for all that satisfy , 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 we can find a neighborhood O of such that holds for all . Since the first definition is easier to work with, we will often use that one.
A function is called continuous at if 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.
- Let . Show that is not continuous at . Hint: Consider the limit along different lines thorough the origin in the complex plane.