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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