Complex Analysis/Complex Functions/Continuous Functions

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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 preimage[edit]

We shall now define and deal with statements of the form

for , , and , 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 functions[edit]

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.