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'.
Example 2.2:
The function
![{\displaystyle f:\mathbb {C} \to \mathbb {C} ,f(z):=z^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91d9da15101e5909cefae27ebeec0ae57e5e9091)
is a complex function.
Limits of complex functions with respect to subsets of the preimage
[edit | edit source]
We shall now define and deal with statements of the form
![{\displaystyle \lim _{z\to z_{0} \atop z\in A}f(z)=w}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3982110bab974979f05434e00327a472ab483f52)
for
,
,
and
, and prove two lemmas about these statements.
Proof: Let
be arbitrary. Since
,
there exists a
such that
.
But since
, we also have
, and thus
,
and therefore
.![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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
.![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
- 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.
Next