Last modified on 17 August 2013, at 16:36

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.

ExercisesEdit

  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