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 $𝔊$ of the complex plane. We then say that the limit of f(z) as z tends to an accumulation point $z_0$ of $𝔊$ exists and equals the complex number L if, for any real number $ϵ>0$ we can find a real number $\delta >0$ such that $|f(z)-L|<ϵ$ for all $z\in 𝔊$ that satisfy $|z-z_0|<\delta$, and we write this limit as

$\underset{z\to z_0}{\lim }f(z)=L$.

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 $z_0$ such that $|f(z)-L|<ϵ$ holds for all $z\in 𝔊\cap O$. Since the first definition is easier to work with, we will often use that one.

A function $w=f(z)$ is called continuous at $z_0$ if $f(z_0)$ is defined and $f(z_0)=\underset{z\to z_0,z\ne z_0}{\lim }f(z)$.

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.

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