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Complex Analysis/Limits and continuity of complex 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'.

Complex functions

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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 preimage

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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 functions

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Definition 2.6:

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

.

Exercises

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  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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