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

$\lim_{z\rightarrow z_0} f(z)=L$.

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 $\epsilon >0$ we can find a neighborhood O of $z_0$ such that $|f(z)-L|<\epsilon$ holds for all $z\in \mathfrak{G}\cap (O\setminus\{z_0\})$. 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 $f(z_0)=\lim_{z\rightarrow z_0} f(z)$.

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.

Exercises

1. Let $f(z)=\frac{(\Re z + \Im z)^2}{|z|^2}$. Show that $f(z)$ is not continuous at $z=0$. Hint: Consider the limit along different lines thorough the origin in the complex plane.

Next