Complex Analysis/Limits and continuity of 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'.

Complex functionsEdit

Definition 2.1:

Let   be sets and   be a function.   is a complex function if and only if  .

Example 2.2:

The function

 

is a complex function.

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 2.3:

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

 ,

we define

 .

Lemma 2.4:

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 2.5:

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

Definition 2.6:

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

 .

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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