# Questions tagged [continuity]

The continuity tag has no usage guidance.

95
questions

**-3**

votes

**1**answer

231 views

### Injective uniformly continuous function $f:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$? [closed]

We say that a function $f: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ is uniformly continuous if there is an integer $K\geq 1$ such that
whenever $(x,y),(x',y')\in \mathbb{Z}\times \mathbb{Z}$ with $|...

**5**

votes

**0**answers

98 views

### “Uniformly continuous” environment sum of a bijection $\varphi:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$

Given any function $f: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ we define the environment sum of $(x,y)\in\mathbb{Z}\times \mathbb{Z}$ with respect to $f$ by
$$\text{es}_f(x,y) = \sum\{f(x', y'): |(...

**1**

vote

**0**answers

140 views

### Reference Request: Stone-Weierstrass on Other Topologies

Let $X$ be a compact subset of $\mathbb{R}^n$. The classical Stone-Weierstrass theorem describes dense subsets of $C(\mathbb{R}^n,\mathbb{R})$ when it is equipped with the compact-open topology.
...

**1**

vote

**0**answers

54 views

### What are the various kinds of graphs that can be defined on $C(X)$

I was considering the space $C(X)$ where $X$ is a topological space and $C(X)$ is the set of all continuous functions from $X$ to $\Bbb R$.
What are the various kinds of graphs that can be defined on ...

**1**

vote

**0**answers

70 views

### Continuity of a constrained parameterized convex optimization problem

Consider the parameterized optimization problem:
\begin{align}
\boldsymbol{s}(p)= &\arg \min_{ \boldsymbol{x}} \quad g( \boldsymbol{x})\\
\text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \...

**-2**

votes

**1**answer

100 views

### Continuity of the Restriction Map Between Function Spaces [closed]

Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as
\begin{align}
\rho:&C(X,Y)\rightarrow C(Z,...

**0**

votes

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53 views

### Continuity of a continuous time martingale

Consider a Brownian motion $(B_t)_{t\ge 0}$ and its natural filtration $\sigma(B_t)$. Suppose $y_t$ is a $[0,1]$ valued, $\sigma(B_t)$ martingale. Does $y_t$ have continuous sample paths? If not, I ...

**2**

votes

**1**answer

72 views

### Continuity of the derivations from semisimple Banach algebras

Let $A$ be a Banach algebra and $X$ a Banach $A$-bimodule. It is known that if $A$ is a $C^*$-algebra, then by Ringrose theorem every derivation $D:A\rightarrow X$ is continuous. Also, a famous ...

**1**

vote

**0**answers

64 views

### Norm closure of $C_b^1(\mathbb{R})$

I want to determine what the closure of $C_b^1(\mathbb{R})$, the space of continuous differentiable functions with bounded derivative, with respect to the supremums norm is. I think that $\overline{...

**1**

vote

**1**answer

97 views

### Continuity of a parameterized convex optimization problem

I have a parameterised optimization problem:
\begin{align}
\boldsymbol{S}(p)= &\arg \min_{ \boldsymbol{x}} g( \boldsymbol{x})\\
\text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \...

**2**

votes

**1**answer

147 views

### Can a bijection between function spaces be continuous if the space's domains are different?

It is well-known that any bijection $\mathbb{R} \rightarrow \mathbb{R}^2$ cannot be continuous. But suppose we have the two spaces $A = \{f(x):\mathbb{R^2}\rightarrow \mathbb{R} \}$ and $B = \{f(x):\...

**0**

votes

**1**answer

182 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

**0**answers

76 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

**1**answer

79 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

**1**answer

123 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 ...