All Questions
46
questions
1
vote
0
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13
views
Continuity of confluent hypergeometric function in terms of its parameters
The confluent hyper geometric function of the first kind (or the Kummer's function) is defined as
$${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a%
\right)}\int_{0}^{1}e^{...
- 3,263
1
vote
0
answers
52
views
Proof that a homeomorphism map boundaries to boundaries
I want to prove that if I have two topological spaces $X$ and $Y$, with $A \subset X$, and a homeomorphism $f : X \to Y$, then $f(\partial A) = \partial \big(f(A)\big)$.
I saw a proof here: https://...
- 1,021
0
votes
0
answers
39
views
Question about proof of Lindelöf Theorem
Supose that $\gamma : [0,1] \to \overline{\mathbb{D}}$ is continuous, $\gamma(t) \in \mathbb{D}$ for $0 \le t < 1$ and $\gamma(1) = 1$. Suppose that $f \in H(\mathbb{D})$ is bounded. If $f(\gamma(t)...
- 1,021
2
votes
1
answer
108
views
Is there a simple way to interpolate smoothly between levels of a complex-valued quadratic map?
I have two complex numbers, $a = x_1 + y_1 i$ and $c = x_2 + y_2 i$. These serve as inputs to a quadratic map $f_n = f_{n - 1}^2 + c$, with $f_0 = a$. Thus the first few iterations of the map are:
$...
- 1,861
0
votes
0
answers
36
views
Identity theorem for (real) analytic functions on lower dimensional subsets
For simplicity, we will deal with $\mathbb{R}^2$. Let's assume we have an one-dimensional submanifold $M_1 \subset \mathbb{R}^2$ and two analytic function $F,G: M_1 \rightarrow \mathbb{R}$.
If I know $...
- 661
2
votes
1
answer
45
views
Continuity of real part of a complex function
Consider a function $g(z)$ which is analytic on $\mathbb{C}_+$, and its range is contained in $\mathbb{C}_-$. Suppose that $g$ has a continuous extension to $\mathbb{C}_+ \cup \mathbb{R}$, denoted by $...
- 115
2
votes
1
answer
73
views
Does the supremum norm $\|p\|_{A}$ depend continuously on subsets $A\subset\mathbb{C}$ with respect to the Hausdorff distance?
Consider the space $\mathcal{K}$ of all non-empty compact subsets of $\mathbb{C}$. One can show that the Hausdorff distance defined by
$$h(X,Y)=\max\bigg\{\sup_{x\in X}\inf_{y\in Y}|x-y|,\sup_{y\in Y}\...
- 3,376
2
votes
0
answers
42
views
A function satisfying a condition is a polynomial of degree $\leq 1$
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function s.t $f(x)=\frac12(f(x+r)+f(x-r))$ for every $r>0, x\in\mathbb{R}$. Prove that $f$ is a polynomial of degree $\leq 1$.
This is a question ...
- 4,653
0
votes
0
answers
38
views
Continuity assumption in uniqueness of primitives (up to a constant)
Stein and Shakarchi asks us to prove that if $f$ is continuous in an open and connected set $\Omega$ that any two primitives of $f$ differ by a constant. I don't understand why the continuity ...
- 1,016
3
votes
1
answer
210
views
Define an operator $T\in B(C[0,1])$ such that: $(Tf)(x)=xf(x)$ for all $x\in [0,1]$ and $f\in C[0,1]$. Prove that $T$ has no eignvalues
Define an operator $T\in B(C[0,1])$ such that:
$(Tf)(x)=xf(x)$ for all $x\in [0,1]$ and $f\in C[0,1]$.
Prove that $T$ has no eignvalues and find $\sigma(T)$.
I think that what is meant is to show that ...
user864806
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
0
votes
1
answer
27
views
Use of uniformity of a limit for proof of continuity when right continuity is known
Consider a function $f: [0,\infty) \rightarrow \mathbb{R}$. Suppose that, for any $t \in (0, \infty)$
$$
\lim_{h \downarrow 0 } \, \, ( f(t+h) - f(t) )= O(h),
$$
namely that the function $f$ is ...
- 200
0
votes
1
answer
94
views
If $g: \mathbb{R}^n \to \mathbb{R}$ then there is a $h$ entire analytic function that extends $ g $
Let $g: \mathbb{R}^n \to \mathbb{R}$ be a continuous function, such that $g(x)>0$, for all $ x \in \mathbb{R}^n$. I want to prove that: there is an entire analytic function $h$ (this means that $...
- 1,657
0
votes
1
answer
698
views
When is the Infinite product of continuous functions a continuous function? Assume that the product is convergent.
Is there any theorem about the continuity of an infinite product of continuous real valued functions on compact Housdorff spaces, if the product is convergent?
I mean, for each natural number $n$, let ...
- 1,544
0
votes
2
answers
41
views
Why does this uniform continuity follow from continuity?
Let $K$ be a compact subset of $\mathbb{R}$. I'm reading a proof and it says the following:
Let $\epsilon > 0$. Since the mapping $\xi \mapsto e^{\imath \, \xi}$
is continuous, we can pick $\...
user745578