All Questions
Tagged with real-analysis gn.general-topology
247 questions
6
votes
0
answers
309
views
Have we discovered constructions for natural fractional dimensional spheres?
I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
- 2,689
4
votes
1
answer
524
views
On the definition of a continuous function
I remember once reading that "a continuous function can be loosely described as a function whose graph can be drawn without lifting the pen from the paper". We all know that this is not true....
- 113
3
votes
2
answers
472
views
Regularity of lipschitz and derivable function
Let be lipschitz $f$ on $[0,1]$ and everywhere derivable. Is it true that $f\in C^1([0,1])$ ?
- 3,312
0
votes
1
answer
243
views
Condition for set of the type $\{(a,b)|a \in A, \ b = f(a)\}$ to have empty interior if $A$ has empty interior [closed]
Let us consider $$\mathcal X = \{(a,b)|a \in A, \ b = f(a)\}, $$ where $A \subset L^1(\mathbb R)$ has empty interior and $f:L^1 \to L^1$ is a bijective map. Does $\mathcal X$ also have empty interior? ...
CommunityBot
- 1
3
votes
1
answer
132
views
Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?
Let $F\subset \Bbb{R}$ intersect every closed uncountable subsets of $\Bbb{R}$.
Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of ...
- 41.8k
0
votes
2
answers
503
views
A Jordan arc in the unit disk
Let $D$ be the open unit disk, and $J$ a Jordan arc (that is, a homeomorphic copy of $[0, 1]$) that lies in $D$, except $J(0)$ lies on the boundary of $D$, say $J(0)=1$. I would like to see that $D\...
- 63.9k
7
votes
2
answers
537
views
How should I understand the "$C^\infty$ functions" whose domain is the dual of $C^\infty(\mathbb{R}^n)$?
I am reading Colombeau's book "New Generalized Functions and Multiplication of Distributions" and he uses the notation $C^\infty({C^\infty}'(\Omega))$ out of nowhere.
Here $\Omega$ is any ...
- 7,817
0
votes
1
answer
290
views
Tensor product is complete?
Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in ...
- 41.1k
1
vote
0
answers
155
views
Study of the class of functions satisfying null-IVP
$\mathcal{N}_u$ : Class of all uncountable Lebesgue-null set i.e all uncountable sets having Lebesgue outer measure $0$.
Let $f:\Bbb{R}\to \Bbb{R}$ be a function with the following property :
$\...
- 307
3
votes
0
answers
74
views
Discreteness of the higher inductive-inductive Cauchy real numbers in real cohesive homotopy type theory
We work in cohesive homotopy type theory with propositional resizing, so that there is only one type of Dedekind real numbers $\mathbb{R}$ up to equivalence, and Mike Shulman's axiom $\mathbb{R}\flat$,...
- 2,624
2
votes
1
answer
148
views
Borel $\sigma$-algebras on paths of bounded variation
Let $(C, \|\cdot\|)$ be the Banach space of continuous paths $x: [0,1]\rightarrow\mathbb{R}^d$ starting at zero with sup-norm $\|\cdot\|$.
Let further $B\subset C$ be the subspace of $0$-started ...
- 463
1
vote
1
answer
183
views
Topological analog of the Lusin-N property
$A\subset \Bbb{R}$ is meager if $A$ can be expressed as a countable union of nowhere dense sets.
Let $f:[a, b]\to \Bbb{R}$ is absolutely continuous, i.e., for every $\epsilon>0$, there exists $\...
- 1,848
3
votes
2
answers
2k
views
Can every real function be approximated with a Riemann-integrable one with any precision required?
Is there some proof that Riemann-integrable functions are dense in the space of all real functions?
In a sense that for every real function $f$ and number $\varepsilon>0$, there is Riemann-...
- 2,821
2
votes
0
answers
73
views
Separately continuous functions of the first Baire class
Problem. Let $X,Y$ be (completely regular) topological spaces such that every separately continuous functions $f:X^2\to\mathbb R$ and $g:Y^2\to \mathbb R$ are of the first Baire class. Is every ...
- 3,317
0
votes
0
answers
94
views
Is the space of affine continuous functions a Baire space
Let $\Omega$ be a compact convex set in q linear normed space. Let $A(\Omega)$ be the space of affine continuous real-valued functions. My question is whether the space $A(\Omega)$ is a Baire space? ...
- 675
3
votes
1
answer
241
views
$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $
I have noticed experimentally that the following question has a positive answer.
Is it true that for all even and convex functions $f$, $g$:
$$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^...
- 128k
3
votes
1
answer
195
views
Positivity of real functions in two variables
Assume that $f_0,f_1,f_2$ are polynomial functions of degree two in two variables. This means that the $f_i$ are linear combinations with real coefficients of $x^2,xy,x,y^2,y,1$.
Consider the function ...
- 128k
19
votes
3
answers
1k
views
Is there a Cantor set $C$ in $\mathbb{R}^{2}$ so the graph of every continuous function $[0,1]\rightarrow [0,1]$ intersects $C$?
Consider the Cantor ternary set on the real line with the usual topology and define a Cantor set to be any topological space $C$ homeomorphic to the Cantor ternary set.
The idea is to construct a ...
- 12.9k
11
votes
3
answers
890
views
Structure theorems for compact sets of rationals
Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in ...
0
votes
0
answers
42
views
Conditions on a set implying properties on neighborhoods
Suppose $F$ is a closed set in a Euclidean space, and for $\epsilon>0$, let $V_\varepsilon$ be the $\varepsilon-$neighborhood of $F$ i.e. the set of points $x$ having a distance less than $\...
- 411
2
votes
0
answers
368
views
Components of the complement of a compact set
Suppose $K$ is a compact subset of $\mathbb{R}^m$ ($m>1$), and $0<r<R$ are fixed numbers. Let $A$ be the set of points having a distance $<R$ and $>r$ from $K$. My questions are
If $K$ ...
CommunityBot
- 1
5
votes
2
answers
646
views
Can functions be differentiable on sets with empty interiors?
As a simple example, suppose we have a function $f: \mathbb{R}^3 \to \mathbb{R}$ defined on the set (and taking $+\infty$ everywhere else),
$$\{x \in \mathbb{R}^3| x_1 \in [-1, 1], x_2 \in [-1, 1], ...
- 351
12
votes
2
answers
678
views
Non-sequential spaces in the wild
TLDR: What are examples of (function-)spaces that are not sequential? When does this matter?
As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...
- 16.4k
5
votes
1
answer
805
views
Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?
I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
- 16.4k
1
vote
0
answers
110
views
Zeroth homology of the complement of a closed set
Suppose $F$ is a closed set in $\mathbb{R}^n$ with $n>1$.
Are there some known conditions that must be imposed on $F$ so that its complement in $\mathbb{R}^n$ has a finite number of components? ...
- 411
155
votes
4
answers
18k
views
Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
- 3,054
18
votes
2
answers
1k
views
Comparing "axiomatized function spaces"
This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context, and note that the existing ...
- 21.2k
3
votes
0
answers
187
views
Analogue of Kolmogorov/Arnold superposition for general manifolds?
Previously asked and bountied at MSE with slightly different language:
Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
- 21.2k
10
votes
1
answer
394
views
Maximally continuous extension of continuous functions from $\mathbb Q$ to $\mathbb R$
Let $f: \mathbb Q \to \mathbb R$ be a continuous function.
An extension of $f$ is a function $\tilde f: \mathbb R \to \mathbb R$ such that $\tilde f = f$ on $\mathbb Q$.
We say an extension $\tilde f$ ...
- 12.5k
30
votes
4
answers
2k
views
is f a polynomial provided that it is "partially" smooth?
Let $f$ be a $C^\infty$ function on $(c,d)$ ,and
let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$.
Suppose for each $n\in ...
- 60.5k
0
votes
0
answers
177
views
On connectedness of the complement
In the application of Runge type theorems on the approximation of functions with some regularity on a neighborhood of a compact, it is interesting to know whether the complement of a compact has ...
- 63.9k
2
votes
1
answer
477
views
Between an open set and its closed subset [closed]
Suppose $F\subset V\subsetneq \mathbb {R}$ where $F$ is closed and $V$ is open. I want to show that $\exists$ an open set $U\subset \mathbb {R}$ satisfying $F\subset U\subset \overline {U}\subset V$. ...
- 128k
8
votes
1
answer
321
views
Must a continuous $\varphi:\mathbb R^n\to\mathbb R^n$ with $\mathbb Q^n \subseteq \varphi[\mathbb Q^n]$ be surjective?
Let $\varphi:\mathbb R^n \to \mathbb R^n$ be just some continuous function.
If the image of $\varphi$ happens to contain $\mathbb Q^n$, does it follow that in fact all of $\mathbb R^n$ is contained in ...
- 9,061
5
votes
0
answers
100
views
What is a mild sufficient condition on $X$ such that $C(X, Y)$ is sequential?
Let $X$ be a topological space, $(Y, d)$ a metric space and $C(X, Y)$ the space of continuous maps with the topology of compact convergence.
Question: What is a minimal topological condition on $X$ ...
- 315
2
votes
0
answers
67
views
A polar open set in a topological subspace?
Suppose $U$ is a bounded open set in $\mathbb{R}^m$ with ($m\geq2$). Is it possible to have a non-empty set $E$ in the boundary $ \partial U$ of $U$ that is open in $ \partial U$ and is polar?
A set $...
- 411
7
votes
1
answer
1k
views
Extending continuous functions from $\mathbb Q$ to $\mathbb R$
Definitions:
Let $E$ be a subset of $X$. By an extension of a function $f: E \to \mathbb R$, I mean a function $\bar f: X \to \mathbb R$ such that $f = \bar f$ on $E$.
Question: For every continuous ...
- 6,155
1
vote
1
answer
267
views
Limit points and Homeomorphism
I was asking this question at Mathematics SE but I got nothing at all. This is why I am trying this site.
We consider the topology of the extended real line. Let $h\colon [-\infty,\infty]\to\Bbb R$ ...
- 1,848
-3
votes
1
answer
315
views
Are the injective functions dense in $C([0,1]^n,\mathbb R^n) $?
Let $n\geq 2$. Are injective functions dense in $C([0,1]^n,\mathbb R^n) $ with the uniform norm?
- 66.3k
1
vote
0
answers
119
views
May sequential continuity of a map on compact sets fail to admit extrema?
Let $X$ be a compact topological space. Is there an example of a sequentially upper-semicontinuous function $f: X \rightarrow \mathbb{R}$ that does not admit a maximum point in $X$?
My very rough ...
- 12.9k
0
votes
1
answer
969
views
Is the pointwise supremum of a continuous function continuous?
Suppose $f(x , y)$ is continuous in both variables. For any $\epsilon > 0$ and some $y_0$, let $h_{\epsilon}(x) = \max_{y^{'}: \| y^{'} - y_0 \| \leq \epsilon} f(x , y^{'})$. It seems to me that $...
- 2,803
0
votes
1
answer
223
views
Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?
Let $X$ be a separable Banach space and $D\subseteq X$ be a
proper, connected, and dense $G_{\delta}$ subset of $X$,
$X-D$ is $\sigma$-porous.
Then is $X-D$ contained in a finite-dimensional ...
CommunityBot
- 1
4
votes
1
answer
121
views
Condition for existence of a continuous function realizing a partition
Let $\{U_i\}_{i=1}^{I}$ be a non-empty and finite collection of non-empty, disjoint, open, (and obviously bounded) subsets of $[0,1]^n$. Suppose also that $[0,1]^n=\cup_{i =1 }^{ I} \overline{U_i}$. ...
0
votes
1
answer
279
views
When does strict inclusion holds for the domain of subdifferential?
Recall that, given an extended real-valued function $f: \mathbb{R}^n \to (-\infty, \infty]$
Its effective domain is,
$$\text{dom}(f) = \{x \in \mathbb{R}^n : f(x) < +\infty\}$$
The subdifferential ...
- 128k
0
votes
1
answer
392
views
Set of zeros of a real function of class $C^1$ [closed]
Let $a < b$ two real numbers and let $f \colon [a,b] \to \mathbb{R}$ a $C^1$ function. Moreover, we consider the set
$$
X := \{ x \in [a,b]\mid f(x) = 0 \}.
$$
Is it the number of connected ...
- 53.3k
5
votes
1
answer
654
views
Fréchet L-Spaces
According to the paper The emergence of open sets, closed sets, and limit points
in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
- 12.7k
0
votes
1
answer
100
views
Connectedness of the set having a fixed distance from a closed set 2
This question is related to this one: Connectedness of the set having a fixed distance from a closed set. Suppose $F$ is a closed and connected set in $\mathbb{R}^n$ ($n>1$). Suppose the complement ...
- 23.2k
-1
votes
1
answer
96
views
Limiting points of elementary set
I consider the following set
$$A:=\left\{ \frac{3mn}{2(m^2+mn+n^2)}; m,n \in \mathbb Z; \text{ and }m,n \text{ are not both zero}\right\}$$
Is it possible to identify the closure of $A$ in the reals?
- 13.1k
11
votes
0
answers
615
views
Is every Baire metric space a complete metric space in disguise?
I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here:
Let's say that a metric space $X$ is Baire if every countable ...
- 63.9k
7
votes
1
answer
2k
views
If $S\subset\mathbb R$ is a $G_\delta$, is there a function $\mathbb R\to\mathbb R$ continuous exactly on $S$?
Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function $\...
- 63.9k
1
vote
0
answers
52
views
A local base for space of probability measures with Prohorov metric
Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
- 111