Questions tagged [continuity]
The continuity tag has no usage guidance.
0
votes
1answer
89 views
Smallest Lipschitz Constant of a Differentiable Function [closed]
Let $X \subset \mathbb{R}^{n}$ be compact and convex. Moreover, let $f:X \rightarrow \mathbb{R}$ be a differentiable map with $\sup_{x \in X} \|\nabla f(x)\| = K < \infty$, where $\|\cdot\|$ ...
1
vote
0answers
54 views
Supremum of an almost surely continuous random process
I was learning this proposition
and now I have a question, how to prove it for an almost surely continuous process? I would be very grateful for any tips.
5
votes
1answer
72 views
continuity of certain map which is defined on a Stonean space
Let $G$ be a discrete group which acts continuously on a Stonean space $\Omega$. Consider the map $f\colon \Omega\to \{0,1\}^G$ sending $x\in \Omega$ to $\chi_{G_x}$, where $\chi_{G_x}$ denotes the ...
0
votes
1answer
92 views
An extension for lower semi continuous lower bounded real valued functions class
Let $(X,d)$ be a complete metric space. I need some explanations about the class of all functions like $f$ which have $f:X \to \mathbb{R}\cup\{ +\infty\}$, be a lower bounded and, for all $y \in X$ we ...
3
votes
1answer
119 views
Continuity concepts for correspondences
Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F ...
1
vote
1answer
84 views
Is continuity preserved under norm operations
Let $F$ be a continuously differenable function over $\mathbb{R}$. Let $\Omega$ be
a bounded subset of $\mathbb{R}^2$. Assume that for every $w\in L^2(\Omega)$ then $v(x)=F(w(x))$, $x\in \Omega$, is ...
3
votes
0answers
60 views
Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?
Recently I came to know about Atsuji space from the paper. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have found ...
0
votes
1answer
42 views
Does $K^{1/2} (t,s)$ inherit the continuity of $K(t,s)$?
Assume that $K(t,s)$ is a (1) symmetric, (2) continuous, and (3) positive definite kernel on $[0,1] \times [0,1]$. The spectral decomposition of $K(t,s)$ is:
$$
K (t,s) = \sum_{i=1}^\infty \lambda_i \...
5
votes
3answers
149 views
Continuity and sequential continuity of a linear functional
Let $E = C_c^0(\mathbb{R}^n;\mathbb{R}^m)$ be the space of compactly supported continuous functions on $\mathbb{R}^n$ with values on $\mathbb{R}^m$. There is a natural norm on this space: given $\...
3
votes
0answers
210 views
Can continuity always be shown by using ε-δ? [closed]
When we learn calculus we usually:
1. Prove that polynomials, the exponential functions, the logarithmic functions, the trigonometric functions, the inverse trigonometric functions are continuous on ...
0
votes
0answers
57 views
Continuity of solution to 2nd Order PDE w.r.t. the coefficients
I am considering the following 2nd order PDE :
On a domain $R$, for some $r > 0$,
\begin{equation*}
\frac{1}{2}\sum_{i, j = 1}^{K}\gamma_{i}\gamma_{j}U_{x_{i}x_{j}}(x) + \sum_{i = 1}^{K}\frac{\...
2
votes
0answers
79 views
Is there a generalization of the Kolmogorov-Chentsov continuity theorem for processes indexed by Banach spaces?
If $(X_t)_{t\ge0}$ is a real-valued stochastic process and for all $T>0$, there are $\alpha,\beta,C>0$ with $$\operatorname E\left[\left|X_s-X_t\right|^\alpha\right]\le C\left|s-t\right|^{1+\...
5
votes
1answer
232 views
affine vs lipschitz
Let $(X,||\cdot||)$ be a normed space where $||\cdot||$ is the sup-norm and let $E$ be a convex and compact subset. Let $f:E\to [0,1]$ be continuous and affine, i.e. for all $x,y\in E$ and all $\...
2
votes
0answers
46 views
$f: \Omega \rightarrow \mathbb{R}$ depending on finite coordinates is $\alpha$-Holder [closed]
Let $M = \{1, \dots, n\}$ be a metric space with the metric $d$ and, in $\Omega = M^{\mathbb{N}}$, define $\tilde{d}(x, y) = \sum_{k=1}^{+\infty} \frac{d(x_k, y_k)}{2^k}$.
We say that $f\colon \...
9
votes
1answer
337 views
Does this Osgood-like condition imply continuity?
Let us consider a bounded, Borel function $F\colon \mathbb R^d \to \mathbb R^d$. Assume it satisfies the following
Osgood-like condition:
$$\tag{O}
\boxed{\vert \langle F(x) - F(y), x-y \rangle\...
2
votes
2answers
220 views
Continuous monotone real functions of several real variables
Let $O$ be an open bounded connected set in $R^n$ and K its boundary. Given a continuous real function $f$ defined on $K$, I would like to extend $f$ to a continuous real function $g$ (i.e. $g$ ...
1
vote
2answers
156 views
Can we get smooth parition of unity with uniformity?
Let $B \subseteq \mathbb{R}^n$ be a product of closed bounded intervals in $\mathbb{R}$. Fix $N>0$. Suppose I want to cover $B$ with $N$ open sets, $U_1, \ldots, U_N$, and get a smooth partition of ...
0
votes
0answers
112 views
Continuity under various topologies for positive linear functionals
It is known that if $\mathcal A$ is a unital $\mathbb C$-$*$-algebra and $A$ is a unital subalgebra closed under $*$, and if $f : A \to \mathbb C$ is linear, then $f$ is positive if and only if $f$ is ...
0
votes
1answer
102 views
Continuity of the solution of a Pde system
Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$ both continuous and bounded.
I have the following system of PDE's
\begin{align}
\begin{cases}
\frac{\partial}{\partial t} u_0(t,r)=- J* ...
7
votes
2answers
226 views
$f$ locally (Lebesgue) integrable function on real line, $g(x):= \lim _{r\to \infty} \frac 1r \int_{x-r}^{x+r} f(t) dt$ exists for every real $x$
Let $f : \mathbb R \to \mathbb R$ be a function such that $f \in L^1[-a,a] , \forall a \in (0,\infty)$ and $g(x) : = \lim _{r\to \infty} \dfrac 1r \int_{x-r}^{x+r} f(t) dt$ exists in $\mathbb R$ for ...
3
votes
0answers
51 views
Sub-quadratic Kolmogorov-Arnold?
The Kolmogorov-Arnold representation theorem says, essentially, that the only multivariate function you really need is addition. (Somewhat) more precisely, it says that for any continuous function $f:...
1
vote
2answers
198 views
How to choose a continuous function which vanishes **only** on the closed set
We are reading John Roe's book Lectures on Coarse Geometry. We come across a question in P27 line 9:
Suppose $X$ is a paracompact and locally compact Hausdorff space, $\bar{X}$ is a ...
3
votes
1answer
117 views
Induced maps on Hyperspace Topologies
If $X$ is a topological space let $2^X$ denote the set of closed subsets. There are multiple topologies one may equip $2^X$ with (in particular, I have in mind the Vietoris, Fell and similar ...
4
votes
0answers
107 views
Does there exist curve (for example, in $\mathbb R^2$) that either touches itself or intersects itself at every one of its points?
I really do not even know how to constructively think about this question that I wanted to post before, but delayed. I know that there are space-filling curves and curves of positive area and those ...
54
votes
19answers
8k views
When has discrete understanding preceded continuous?
From my limited perspective, it appears that the understanding
of a mathematical phenomenon has usually been achieved,
historically, in a continuous setting
before it was fully explored in a discrete ...
1
vote
0answers
102 views
Continuity of a convex function on a vector bundle
Consider the rank-${n \choose m}$ vector bundle $\pi\colon E:=\bigwedge^m(TN)\to N$ over a smooth Finsler manifold $N$ and equip each fibre $E_q := \pi^{-1}(q)$ with a norm that depends smoothly on $q\...
11
votes
2answers
365 views
Brouwer's Theorem in the free topos?
In Introduction to Higher-Order Categorical Logic, Lambek & Scott remark that Brouwer's Theorem (all functions $\mathbb{R}\to\mathbb{R}$ are continuous) holds in the free topos $\mathcal{T}$.
...
15
votes
2answers
420 views
Does the class of Hausdorff spaces have a shared “Coordinate space”?
Given topological spaces $X$ and $C$ we call $C$ a coordinate space for $X$ to mean that every open set $U \subset X$ is of the form $f^{-1}(V)$ for some open $V \subset C$ and continuous $f \colon X \...
3
votes
1answer
96 views
Density of the max set of a non-differentiable function
For $f : [0;1] \to \mathbb{R}$, let $M_f := \{x \in [0;1] \mid f(x)$ is a local strict maximum of $f\}$. It is easy to see that for any $f$, $M_f$ is at most countable. It is also easy to see that ...
7
votes
2answers
415 views
Continuous functions and infinity
Suppose $f(x)$ is continuous on $\mathbb{R}$, for $\forall \delta>0, \forall x\in\mathbb{R}, \lim_{n\rightarrow\infty}f(x+n\delta)=+\infty$. Is it correct that $\lim_{x\rightarrow+\infty}f(x)=+\...
1
vote
1answer
429 views
properties of orderd upper and lower semi continuous functions [closed]
$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$.
If $f(x_0) = g(x_0) $ for some point $x_0\in M$,
is it ...
2
votes
0answers
123 views
Left multiplication Fréchet Differentiable
Let $L^2$ be a separable Hilbert space and let $\{e_i(t)\}_{i \in \mathbb{N}}$ be a basis for it. Moreover let $\phi(T,z)$ and $\psi(T,z)$ be 1-parameter families (in z) in $L^2$, with basis ...
5
votes
2answers
903 views
Upper semicontinuity of set-valued maps with open values
Let $X$ and $Y$ be metric spaces. The $(\varepsilon,\delta)$-definition of continuity of single-valued maps can be rephrased as:
Let $f$ be a single-valued map from $X$ to $Y$. $f$ is continuous at ...
32
votes
2answers
923 views
can another topology be given to $\mathbb R$ so it has the same continuous maps $\mathbb R\rightarrow \mathbb R$?
We say two topologies $\tau$ and $\rho$ on $X$ are similar if the set of continuous functions $f:(X,\tau) \rightarrow (X,\tau)$ is the same as the set of continuous functions $f:(X,\rho)\rightarrow (X,...
3
votes
0answers
196 views
Points of continuity of a lower semicontinuous function have non empty interior
Let $X\subset \mathbb R^d$ having non empty interior and let $f:X\to\mathbb{R}$ be lower semicontinuous.
I know that the set of discontinuities of such a function is contained in a meager set, and ...
3
votes
1answer
520 views
A continuity/bootstrap argument
I am trying to understand how one can prove the following assertion using a continuity argument:
Let $0<\epsilon<\epsilon_0$. Let $I=[t_0,R]$ be a compact interval. Suppose that $S:I\to [0,\...
2
votes
1answer
120 views
Modulus of Continuity for an Analytic Function on an Ellipse
Given $f\in C^{\infty} (E)$, where $E\subseteq \mathbb{C}$, define $E_{\rho} \subseteq \mathbb{C}$ as the maximal ellipse with foci at $\{-1,1\}$ where $f$ is analytic, and semi-minor + semi-major ...
1
vote
1answer
173 views
$L^\infty(0,T;X) \cap C([0,T];Y) \subset C([0,T];X)$ for $X \subset Y$ dense?
is the Inclusion stated in the title true? In my case the spaces (essentially) are $X = H^1(\Omega)$ and $Y = L^2(\Omega)$ for $\Omega \subset \mathbb{R}$ bounded. My first try was to show
$\lim_{t_1 ...
4
votes
1answer
229 views
Continuous non-constant function with infinite intersections with horizontal line on a compact interval?
The title might be misleading, but whether such a function exists is what boggles me about the following problem:
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that for all $...
1
vote
0answers
81 views
Is support function of a convex curve in $\mathbb{R}^2$ absolutely continuous? [closed]
There is an example of a convex function of two variables that is convex but not absolutely continuous. The level set of this function is a convex curve.
To construct one example of such a function ...
6
votes
1answer
345 views
Finding non convex functions satisfying a weak form of convexity, without the axiom of choice
If a real-valued function $f$ over reals satisfies $$ (1) \; \; \; f({x+y\over2})\le {f(x)+f(y)\over2}, $$and it is continuous, then it is not hard to see that $f$ is indeed convex. On the other hand,...
5
votes
1answer
207 views
Is the heat kernel for the hyperbolic plane uniformly continuous in $t\in(0,\infty)$?
let $\mathbb{H}$ be the hyperbolic plane and let $k(t,x,y)$ be the associated heat kernel.
I am wondering, if for any fixed $y\in M$ and $\epsilon >0$ the function $u_t(x):=k(t,x,y)$ is continuous ...
20
votes
3answers
519 views
Which partitions of $[0,1]$ are collection of level sets of a real continuous function?
Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...
7
votes
0answers
178 views
Uniform approximation of separately continuous functions on zero-dimensional spaces
For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...
1
vote
0answers
59 views
Extending an homotopy, knowing the two base functions extend
Let $A\subset B$ be paracompact spaces, and let $C$ be a paracompact space.
Let $f_0,f_1:A\rightarrow C$ be continuous functions. $F:A\times[0,1]\rightarrow C$ a homotopy from $f_0$ to $f_1$. Suppose ...
0
votes
0answers
89 views
Convergence of the solution of Volterra integral equation with convergent kernel (reposted, need help!)
Consider the following Volterra integral equation
$g(t)=∫_0^tK_n(t,s)w_n(s)ds$
where $g(t)$ and $K_n(t,s)$ are continuous and $K_n(t,s)≥K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ converges to ...
1
vote
3answers
354 views
Topological properties for which bijectively related imply homeomorphism
In this post I give examples of topological spaces for which bijectively relations imply existence of an homeomorphism. Namely:
Intervals of the real line.
Compact spaces.
I also give a ...
2
votes
0answers
88 views
On compactness in $C(X)$
Let $X$ be a Tychonoff space. It is well known, that for a family of scalar functions equicontinuity + pointwise boundedness imply relative compactness in $C(X)$ (with compact-open topology). It is ...
3
votes
2answers
378 views
Is a sigma-finite Borel measure over $\mathbb R$ determined by its values on the continuous functions? [closed]
Suppose that $\mu$ and $\nu$ are sigma-finite measures on the Borel sigma-algebra over $\mathbb R$ such that $\int_{\mathbb R}f\,d\mu=\int_{\mathbb R}f\,d\nu$ for all nonnegative continuous functions $...
3
votes
1answer
135 views
If $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, does $f(u_n) \to f(u)$ in $H^{\frac 12}(\partial\Omega)$ for $f$ Lipschitz?
Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with $|f'(x)| \leq C$ for all $x$ and $f(0)=0$.
Suppose $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, where $\Omega$ is a bounded domain of ...
