All Questions
23
questions
1
vote
0
answers
52
views
Proof that a homeomorphism map boundaries to boundaries
I want to prove that if I have two topological spaces $X$ and $Y$, with $A \subset X$, and a homeomorphism $f : X \to Y$, then $f(\partial A) = \partial \big(f(A)\big)$.
I saw a proof here: https://...
- 1,021
1
vote
0
answers
44
views
Understanding continuity in $\hat{\mathbb{C}}$
Let $\hat{\mathbb{C}}$ denote the Riemann sphere. Let $f:B_1(0) \to \hat{\mathbb{C}}$ be continuous. If $f$ is continuous at $z$ and non-zero, then $1/f(z)$ is continuous at $z$ as well. My question ...
- 730
0
votes
0
answers
36
views
Identity theorem for (real) analytic functions on lower dimensional subsets
For simplicity, we will deal with $\mathbb{R}^2$. Let's assume we have an one-dimensional submanifold $M_1 \subset \mathbb{R}^2$ and two analytic function $F,G: M_1 \rightarrow \mathbb{R}$.
If I know $...
- 661
0
votes
1
answer
332
views
definition of limits and continuity in complex analysis
Here is the definition my textbook gives:
Suppose a have a function $f$ with domain $\{z \in \mathbb{C} : |z| \le 1\}$. The point $i$ has $|i|=1$ and is in $f$'s domain. $f$ is not defined on any ...
- 77
0
votes
1
answer
113
views
Show that if $M ≥ 0$ and $|f(z)| ≤ M$ for all $z ∈ ∂V$ , then $|f(z)| ≤ M$ for all $z ∈ V $
Suppose that V is a bounded open subset of the plane and $f ∈ C(\overline V)
∩ H(V)$ i.e. $f$ is continuous on $\overline V$ and $f\restriction_V$ is holomorphic on $V.$
Show that if $M ≥ 0$ and $|f(z)...
- 61
6
votes
2
answers
2k
views
No continuous injective functions from $\mathbb{R}^2$ to $\mathbb{R}$
Which of the following statements is true?
$(a)$ There are at most countably many continuous maps from $\mathbb{R}^2$ to $\mathbb{R}$
$(b)$ There are at most finitely many continuous surjective maps ...
- 2,589
-2
votes
1
answer
77
views
Continuous bijective mapping between $\mathbb{C}$ and $\mathbb{C}^*$
Is there a continuous bijective mapping from $\mathbb{C}$ to $\mathbb{C}^* = \mathbb{C}\setminus\{0\}$?
4
votes
1
answer
876
views
When does a bounded continuous function extend continuously to its closure
Let $\Omega$ be a domain in $\mathbb{C}^n$. Let $f:\Omega\longrightarrow\mathbb{C}$ be a bounded continuous function. I wanted know if there are any necessary and sufficient conditions for $f$ to be ...
- 386
0
votes
2
answers
304
views
Proving a function to be an open map
How can we prove that the function $f: S^1 \rightarrow S^1$ defined as $z \mapsto z^2$ is a continuous and open map using topological arguments? Here $S^1$ represents the unit circle in complex plane, ...
- 350
1
vote
2
answers
251
views
Closure and continuous map
Let $S_i$ be a countable family of sets and $f$ a continuous map (say everything is in the complex plane).
Is it true that
$$
f( \overline{ \cup_i S_i } ) =
\overline{ \cup_i f( S_i ) }
$$
or perhaps ...
- 3,646
2
votes
0
answers
443
views
Separate continuity and analyticity in one variable implies joint continuity?
A theorem of Hartog states that if $U \subseteq \mathbb C$ is open and $f : U \times U \to \mathbb C$ is analytic when we fix any variable ('separately analytic'), then it is continuous.
In general, ...
- 26.5k
0
votes
1
answer
82
views
How $f(z)=z^n$ is continuous but its inverse doesn't map an open set to an open set?
If a map $f: X \to Y$ is continuous then $f^{-1}(V)$ is open set in $X$ whenever $V$ is an open set in $Y$.
Consider $f: \mathbb{C} \to \mathbb{C}$ by $f(z)=z^n$. By this map the sector $r≤a$, $0≤θ&...
user231343
0
votes
2
answers
71
views
$\mathbb{C}$ similar to $\mathbb{R^2}$ for some properties
I have a doubt that I would like to clear, it concerns the similarities between $\mathbb{R^2}$ and $\mathbb{C}$.
In particular, I am investigating the properties about the compactness of the domain ...
- 584
0
votes
3
answers
4k
views
Does a continuous mapping have to map the boundary to the boundary
This question comes from when I read my Complex Analysis book.
On the chapter of Mobius transformation, there is an example:"construct a Mobius transformation that maps $\{\mathrm{Im} (z)>0\}$ ...
- 1,856