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
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We shall now define and deal with statements of the form
![{\displaystyle \lim _{z\to z_{0} \atop z\in B'}f(z)=w}](https://wikimedia.org/api/rest_v1/media/math/render/svg/175360f4d902a363b3f0b01d80564b4e5f35d120)
for
, and prove two lemmas about these statements.
Proof: Let
be arbitrary. Since
![{\displaystyle \lim _{z\to z_{0} \atop z\in B'}f(z)=w}](https://wikimedia.org/api/rest_v1/media/math/render/svg/175360f4d902a363b3f0b01d80564b4e5f35d120)
there exists a
such that
![{\displaystyle z\in B'\cap B(z_{0},\delta )\Rightarrow |f(z)-w|<\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/22a278bafe80653928aef7ac029567d336791739)
But since
, we also have
, and thus
![{\displaystyle z\in B''\cap B(z_{0},\delta )\Rightarrow z\in B'\cap B(z_{0},\delta )\Rightarrow |f(z)-w|<\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d09a5be54b091b039d81c678e7450ed229f0ddeb)
and therefore
![{\displaystyle \lim _{z\to z_{0} \atop z\in B''}f(z)=w}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f17ae5ba0233a5180cac6ceb00a9fc06c51052e)
![{\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
![{\displaystyle \lim _{z\to z_{0} \atop z\in O}f(z)=w}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d86da448fe1d1bc0efd3ce7c31514be2b1d52083)
there exists a
such that:
![{\displaystyle z\in B(z_{0},\delta _{2})\cap U\Rightarrow |f(z)-f(z_{0})|<\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3aabb3905ac59e4d1505e728cda5fc7ead56ca76)
We define
and obtain:
![{\displaystyle z\in B(z_{0},\delta )\cap B'\Rightarrow z\in B(z_{0},\delta )\Rightarrow z\in B(z_{0},\delta _{2})\cap U\Rightarrow |f(z)-f(z_{0})|<\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/901a0aef8a556a480da3ad1fb47c21f3845eff33)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
We recall that a function
![{\displaystyle f:M\to M'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d43916c27b86573fbc804fcdc6e57fba6dbaf6)
where
are metric spaces, is continuous if and only if
![{\displaystyle x_{l}\to x,l\to \infty \Rightarrow f(x_{l})\to f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/157385cb2108431ba16ad04c36002c82852d6417)
for all convergent sequences
in
.
- Proof
- Prove that if we define
![{\displaystyle f:\mathbb {C} \to \mathbb {C} ,f(z)={\begin{cases}{\dfrac {z^{2}}{|z|^{2}}}&:z\neq 0\\1&:z=0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/deb1c25ea244d0bbdb29dd54525e94d4e4b871d4)
- 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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