# Complex Analysis/Complex Functions/Continuous Functions

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 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.

## Exercises[edit]

- Let . Show that is not continuous at . Hint: Consider the limit along different lines thorough the origin in the complex plane.