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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 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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