Complex Analysis/Complex Functions/Continuous Functions

In this section, we

  • introduce a 'broader class of limits' than known from real analysis (namely limits with respect to a subset of ) and
  • characterise continuity of functions mapping from a subset of the complex numbers to the complex numbers using this 'class of limits'.

Limits of complex functions with respect to subsets of the preimageEdit

We shall now define and deal with statements of the form


for   , and prove two lemmas about these statements.

Definition 2.2.1:

Let   be a set, let   be a function, let   , let   and let   . If


we define:


Lemma 2.2.2:

Let   be a set, let   be a function, let   , let   and   . If




Proof: Let   be arbitrary. Since


there exists a   such that


But since   , we also have   , and thus


and therefore


Lemma 2.2.3:

Let   ,   be a function,   be open,   and   . If


then for all   such that   :


Let   such that   .

First, since   is open, we may choose   such that   .

Let now   be arbitrary. As


there exists a   such that:


We define   and obtain:


Continuity of complex functionsEdit

We recall that a function


where   are metric spaces, is continuous if and only if


for all convergent sequences   in   .

Theorem 2.2.4:

Let   and   be a function. Then   is continuous if and only if



  1. Prove that if we define
    then   is not continuous at   . Hint: Consider the limit with respect to different lines through   and use theorem 2.2.4.