# Complex Analysis/Complex Functions/Continuous 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 preimage[edit]

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 functions[edit]

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**:

## Exercises[edit]

- Prove that if we define