Questions tagged [continuity]

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Convergence in expectation of a discontinuous function

Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, ...
dhp's user avatar
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Homeomorphic extension of a discrete function

Let $f : \{ 0,1 \} ^ {n} \rightarrow \{ 0,1 \} ^ {n}$ be a bijective map. Then is there a known computable way to extend it to a homeomorphism $g:[ 0,1 ] ^ {n} \rightarrow [ 0,1 ] ^ {n}?$
Guill Guill's user avatar
5 votes
1 answer
183 views

On the continuity of a Set-Valued function (correspondence) [closed]

Let $f:\mathbb{R}^{n}\rightrightarrows \mathbb{R}^{m}$ be a set-valued function defined by \begin{equation*} f\left( x\right) =\left\{ y\in \mathbb{R}^{m}:g\left( x\right) +h\left( x\right) ^{T}y\...
UnclePetros's user avatar
7 votes
1 answer
457 views

Non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections between them in both sides

I need to construct an example of two non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections between them in both sides. Spaces should have induced ("good&...
jkjfgk's user avatar
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2 answers
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Asymptotics of the unique root of a polynomial equation defined as a sum of rational expressions

Let $\lambda_1\ge \ldots \ge \lambda_n \gt 0$. Define a function $F:\mathbb R_+ \to \mathbb R_+$ by $$ F(t) = t^2\sum_{i=1}^n\frac{\lambda_i^2}{(\lambda_i + t)^2}. $$ It is clear that $F$ is strictly ...
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1 answer
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Constructing a Gaussian process on $[0, 1]$ such that the sample paths are $1$-Lipschitz continuous with high probability?

In the paper [1] the authors demonstrate that for a centered Gaussian process $\{X_t\}_{t \in [0, 1]}$, if there is a constant $C > 0$ such that $$ \mathbb{E}[(X_t - X_s)^2] \leq C~(t- s)^2, $$ ...
Drew Brady's user avatar
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Under which conditions Mean Square Continuity implies Sample Continuity for Gaussian Processes?

First, let us give the setting. Let $(\Omega, \Sigma, \mathbf{P})$ be a probability space, let $T$ be some interval of time, and let $X: T \times \Omega \rightarrow S$ be a stochastic process. By Mean ...
Grandes Jorasses's user avatar
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0 answers
75 views

How to show two equivalent projection in a C∗ algebra are not homotopic

Show that two equivalent projections need not be homotopic. HINT: Let $P=\begin{pmatrix} 1&0\\0&0\end{pmatrix}$ and $Q=\begin{pmatrix} t&\sqrt{t(1-t)}\\\sqrt{t(1-t)}&1-t \...
GBA's user avatar
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Uniformly open map on a dense subset

Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion. I think the ...
user243245's user avatar
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On "canonical" extensions of functions from integers to reals

Although this is essentially a port of my MathSE question, I think the users there tend to not understand how to interpret the questions from a higher perspective (and often too literally). This is ...
Graviton's user avatar
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Continuous dependence of the (infinite) roots of a polynomial on its coefficients

I'm trying to show the continuous dependence of the roots of a polynomial on its coefficients when the root number can be infinite (e.g., $x-y$). I don't know much about algebraic geometry but after I ...
Kryvtsov's user avatar
2 votes
1 answer
123 views

A characterization of continuity in terms of preservation of connected sets. Where to find the result?

There is a result that if $X$ is a locally connected space and $Y$ is a locally compact Hausdorff space, then a function $f \colon X \to Y$ is continuous if and only if $f$ has a closed graph and for ...
Calvin Chin's user avatar
2 votes
1 answer
377 views

(Dis)prove : if every function with closed graph are continuous then the target space is compact

$(X, \tau_X) $ and $(Y, \tau_Y) $ be two topological spaces. $\forall f\in Y^X$ with $\text{Gr}(f) $ is closed implies $f\in C(X, Y) $. Question : Does this implies $(Y, \tau_Y) $ is compact? ...
Sourav Ghosh's user avatar
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What can be said about cluster sets for power series of two variables?

I'm still trying to prove the continuity of a function $u$ which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ...
Raphael B's user avatar
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Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?

Say we have a power series of two variables, with an associated function $f$ defined as $$ \begin{split} f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\ & a_{n,m} \geq 0 \quad \forall n, m \in\...
Raphael B's user avatar
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83 views

From convergence pointwise to convergence of the supremum for semicontinuous functions

Let $K\subset\mathbb{R}$ a compact set, and $(f_n)_{n\geq 1}$ and $f$ upper semicontinuous functions over $K$ (taking hence values in $\mathbb{R}\cup\{-\infty,+\infty\}$) such that for all $x\in K$, ...
G. Panel's user avatar
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Commutation between $\sup$ and $\lim$ for a decreasing sequence of functions

Let $S_d$ be the relative interior of the $d$-simplex: $$S_d=\big\{x\in\mathbb{R}^d,\,\forall 1\leq i\leq d,\,x_i>0,\,\sum_{i=1}^d x_i=1\big\}$$ and $g\in\mathcal{C}^0(S_d,\mathbb{R})$ such that $\...
G. Panel's user avatar
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Discontinuity of the Fourier transform of $ x \mapsto (1+ x^2)^{- \gamma/2}$ for $\gamma \leq 1$

Fix $\gamma > 0$. Let $\mathcal{F}$ be the Fourier transform and consider the function $f(x) = (1+ x^2)^{- \gamma/2}$ for $x \in \mathbb{R}$. This function is in $\mathcal{S}'(\mathbb{R})$ and its ...
Goulifet's user avatar
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2 votes
0 answers
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Continuous analogue for Szpilrajn Theorem: complete preorder extends a continuous preorder

A corollary of Szpilrahn Theorem states: Any preorder on nonempty $X$ has a complete and transitive extension. I am thinking about the "Szpilrahn Theorem" for continuous preorder on ...
dodo's user avatar
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1 answer
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Maps that preserve winding numbers

This question is a cross post from the Math StackExchange since it got no attention at all there: https://math.stackexchange.com/questions/4414601/maps-that-preserve-winding-numbers I am looking for a ...
Manuel Eberl's user avatar
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3 votes
1 answer
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Is a convex, lower semicontinuous function that is bounded from below, actually continuous?

While thinking about convex functions, I managed to put together the following proof which I find a bit too good to be true. $X$ is a topological vector space that is also a Baire space. Lemma: Let $f ...
iolo's user avatar
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Is $g = \sum_{n \in \mathbb{Z}} f(\cdot - n)$ continuous if $f$ is vanishing, continuous, and integrable?

Let $f \in \mathcal{C}_0(\mathbb{R}) \cap L^1(\mathbb{R})$ be a continuous and integrable function such that $f(x) \rightarrow 0$ when $|x|\rightarrow \infty$. The sequence of a functions $f_N = \sum_{...
Goulifet's user avatar
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2 votes
0 answers
194 views

Convolution of continuous compactly supported functions on étale groupoid is continuous

Let $G$ be an étale Hausdorff groupoid, i.e. a topological groupoid $G$ such that the source and range maps $s,r: G \to G$ are local homeomorphisms. Consider the complex vector space $C_c(G)$ of ...
Andromeda's user avatar
3 votes
2 answers
290 views

Smoothing a map $f:X\to \mathbb{R}$ while fixing it over a closed $C\subset X$

$\newcommand{\R}{\mathbb{R}}$I have a map $f\in C^0(X,\mathbb{R})$, where $X$ is a compact and Hausdorff topological space, which is a manifold outside of a compact subset $K\subset X$. I would like ...
Overflowian's user avatar
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5 votes
1 answer
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Is $\mathbb{Q}$ the orbit of a continuous function that is computable when restricted to $\mathbb{Q}$?

In the previous post What is the smallest set of real continuous functions generating all rational numbers by iteration? I asked for the smallest set of continuous real functions that could generate $\...
Ivan Meir's user avatar
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1 vote
1 answer
96 views

Given an increasing function, need to construct a continuous increasing function equivalent to given function

Given an increasing function $f:[0,\infty)\to[0,\infty)$, we can define $$F(x)=\int_x^{x+1} f(t)dt,$$ which is continuous, increasing function satisfying $$f(x)\leq F(x)\leq f(x+1).$$ Question) For ...
Wilderness's user avatar
2 votes
0 answers
84 views

Can a smooth function always fit between two non-smooth functions? [closed]

Suppose I have two continuous functions, $f$ and $g$, with $f(x)<g(x)$ for all $x$ in some closed domain. Is it always possible to find a (piecewise, if needed) smooth function $h$ such that $f(x)&...
Sam Benner's user avatar
9 votes
2 answers
597 views

Is $\mathbb{Q}$ the orbit of a rational function under iteration?

In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous function suffices. In the ...
Ivan Meir's user avatar
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3 votes
0 answers
66 views

Is there a finite set of polynomials generating all rational numbers by iteration?

In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb{Q}$ by iteration starting from 0. Surprisingly one continuous function suffices. The ...
Ivan Meir's user avatar
  • 4,552
33 votes
2 answers
2k views

What is the smallest set of real continuous functions generating all rational numbers by iteration?

I recently came across this problem from USAMO 2005: "A calculator is broken so that the only keys that still work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The ...
Ivan Meir's user avatar
  • 4,552
2 votes
1 answer
171 views

Relationship between $C(X\times Y,Z)$ and $C(X,C(Y,Z))$

Let $X$, $Y$, and $Z$ be locally-compact, complete, and separable metric spaces and suppose that $X$ is compact; all non-empty. Consider the spaces $C(X,C(Y,Z))$ and $C(X\times Y,Z)$ both equipped ...
SetValued_Michael's user avatar
2 votes
0 answers
92 views

Are a.e. derivatives of continuous $VBG_*$ functions Denjoy–Perron integrable?

I would like to ask a question pertaining to the Denjoy–Perron (Henstock–Kurzweil) theory of integration. It is simple enough that I have entertained the idea that perhaps an answer is known, but I ...
David Manolis's user avatar
3 votes
0 answers
166 views

Is the following generalization of piecewise continuity equivalent to any other common types of functions on metric spaces?

EDIT: I think what I have isn't precisely what I want... we should also require $x$ in condition (3) to be "not bad" in some sense, although I'm not quite sure what that should mean for my ...
exfret's user avatar
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0 votes
1 answer
236 views

Approximation of positive right-continuous function

Let $f:(0, +\infty)\to(0, +\infty) $ be a monotone decreasing, right-continuous function. Can we find a sequence $\{f_{n}\}_{n\in \mathbb{N}}$ of strictly monotone decreasing, continuous functions, ...
Shaq155's user avatar
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1 vote
0 answers
60 views

Integrating over correspondences

Consider two compact sets $X$ and $Y$, a function $f:Y\to \mathbb{R}$, and a closed, non-empty correspondence $A:X\twoheadrightarrow Y$. Define the function $G:X\to \mathbb{R}$ by $$ G(x)=\int_{A(x)}f(...
tsm's user avatar
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1 vote
1 answer
1k views

Norms in Sobolev space $W^{1,\infty}$

Let $n\in\mathbb{N}$ and consider the Sobolev space $W^{1,\infty}(\mathbb{R}^n)=\lbrace u\in L^{\infty}(\mathbb{R}^n):\partial_iu\in L^{\infty}(\mathbb{R}^n) \rbrace$. A function is in $W^{1,\infty}$ ...
Oliver Watt's user avatar
3 votes
1 answer
327 views

Functions with at most linear growth at infinity: is the constant itself continuous?

I am considering the family $\mathcal{F}$ of functions $f \colon \mathbb{R} \to \mathbb{R}$ which have at most linear growth at infinity, that is there exists a constant $M_f$ such that \begin{...
A. Pesare's user avatar
  • 172
-4 votes
1 answer
188 views

strict convexity and Lipschitz continuity [closed]

Consider a continuously differentiable function $f: \mathbb{R}^n \mapsto \mathbb{R}$. If $f$ is strictly convex, does it imply that it is not Lipschitz on $\mathbb{R}^n$? Because if $f$ is strictly ...
Trb2's user avatar
  • 25
3 votes
1 answer
717 views

Function whose sets of discontinuities and zeros are the rationals

Question: Is there a real valued function $f:\mathbb{R}\to\mathbb{R}$ such that its set of discontinuities is $\mathbb{Q}$ and its set of zeros $\{x\in \mathbb{R}\mid f(x)=0\}$ is also $\mathbb{Q}$? ...
tj_'s user avatar
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-1 votes
1 answer
108 views

Sobolev injections [closed]

It is true to write that $W^{1,\infty}(]0,\infty[) \hookrightarrow C([0,\infty[)$ et $W^{1,1}(]0,\infty[) \hookrightarrow C([0,\infty[)$ ? Thanks
user895874's user avatar
0 votes
1 answer
557 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 $...
Saeed's user avatar
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3 votes
1 answer
224 views

Is disintegration continuous?

Let $X,Y$ be Polish spaces and suppose that $X$ is compact. Denote by $\mathcal{Mes}(X,\mathcal{P}(X\times Y))$ the set of (Borel) measurable functions from $X$ to the set of Borel probability ...
ABIM's user avatar
  • 4,956
2 votes
1 answer
150 views

Largest number N for which injective mapping $f: 2^N \to 2^8 \times 2^8 \times 2^8$ which is Lipschitz-1 CT with $K\leq 3$ exists

I have a function on $h: [0,1] \to [0,1]$ whose output is smooth (polynomial of low degree), and I need to discretize it but I need to save it with three 8 bit numbers. These three 8 bit numbers need ...
polpetti's user avatar
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1 vote
0 answers
30 views

How to prove continuity in topological group action of ${\rm{GA}}_a(X)$ on $T(X)$, to make ${\rm{GA}}(X)$ a topological group?

The question comes from the following paragraph of a text on geometry in the context of affine geometry (Marcel Berger et al., "Geometry I", P56-57): 2.7.1.3. If we don't want to resort to ...
zzzhhh's user avatar
  • 111
-2 votes
1 answer
318 views

Is a function piecewise continuous if it has a left-limit and a right-limit at every point in its interval domain and equals at least one of these? [closed]

Suppose a real-valued function f, whose domain is an interval, has the property that at every point in its domain it has a left-limit and a right-limit, and it equals at least one of these. Is it ...
immeasurable's user avatar
5 votes
1 answer
282 views

Points of differentiability of squared distance from a point in metric spaces

I posted this same question on MSE with no answer. Let $I:=(0, + \infty)$ and let $(X,d)$ be a complete and separable metric space. In this setting we say that $u : I \to X$ is absolutely continuous ...
Bremen000's user avatar
  • 327
3 votes
1 answer
68 views

Analyticity of $f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$ in the complex plane?

Let I have the following function, $f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$ Where, $x,y \in C$, $a,b\in R$ and $- m \le \Re (x),\Re (y),\Im (x),\Im (y)...
Samantha's user avatar
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1 vote
0 answers
827 views

Weak sequential continuity vs strong continuity

Let $E$ be a Banach space, $T:E\rightarrow E$ a non-linear operator. $T$ is said to be Weakly Sequentially Continuous (shortly W.S.C) on $E,$ if for every $\left(x_{n}\right)_{n}\subseteq E$ with $x_{...
Motaka's user avatar
  • 291
0 votes
2 answers
284 views

Is the restriction map $C^1\ni f\mapsto\left.f\right|_K$ a continuous map?

Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $\iota$ be a continuous embedding of $\Theta$ into $C^1(E,E)$. I would like to show that, ...
0xbadf00d's user avatar
1 vote
0 answers
589 views

A weakly sequentially continuous operator which is not weakly continuous

I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity. So, let $T$ an operator between a Banach space $X$ and itself. $T$ is weakly ...
Motaka's user avatar
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