Complex Analysis/Complex Functions/Continuous Functions

< Complex Analysis‎ | Complex 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 , 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

then

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 :

Proof:

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

Proof:

ExercisesEdit

  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.

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