# 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, $x=a$, ...
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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 $g:[ 0,1 ] ^ {n} \rightarrow [ 0,1 ] ^ {n}?$
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### 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\...
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&...
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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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### Constructing a Gaussian process on $[0, 1]$ such that the sample paths are $1$-Lipschitz continuous with high probability?

In the paper  the authors demonstrate that for a centered Gaussian process $\{X_t\}_{t \in [0, 1]}$, if there is a constant $C > 0$ such that $$\mathbb{E}[(X_t - X_s)^2] \leq C~(t- s)^2,$$ ...
61 views

### 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 ...
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### 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 ...
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 ...
1 vote
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### 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 ...
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 ...
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 ...
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### 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, ...
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 ...