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 z₀ of $\mathfrak{G}$ exists and equals the complex number L if, for any real number ε > 0 we can find a real number δ > 0 such that $|f(z)-L|<\epsilon$ for all $z \in \mathfrak{G}$ that satisfy 0 < | z − z₀ | < δ, 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 ε > 0 we can find a neighborhood O of z₀ 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 z₀ if f(z₀) 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.

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