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        Questions tagged [continuity]

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        2
        votes
        2answers
        102 views

        Supremum of continuous functions and essential supremum of continuous functions

        Suppose that $(X,d)$ is a Polish metric space and $A$ is a set of continuous bounded functions $f:X\to \mathbb{R}$. Suppose that $\mu:X\to[0,1]$ is a Borel probability measure. Define $$\sup A:X\to ...
        1
        vote
        1answer
        155 views

        Continuity of subharmonic functions

        There is a result saying that the set where a subharmonic function defined on an open set of $\mathbb{R}^{m}$ ($m\geq2$) is discontinuous is a polar set. Could someone give me a reference for this ...
        0
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        0answers
        74 views

        Maximum theorem with linear constraints. On parametric continuity of in optimization

        Given \begin{align} s(\theta)= &\text{arg min}( g( \boldsymbol{x}) ) \\ \text{subject to }& \boldsymbol{A}(\theta) \boldsymbol{x} = \boldsymbol{b}(\theta) \\ &c_1 \le x_i \le c_2 , ...
        -2
        votes
        1answer
        263 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
        1answer
        180 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
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        0answers
        144 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
        0answers
        57 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
        0answers
        80 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
        1answer
        101 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
        0answers
        55 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
        1answer
        80 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
        0answers
        65 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
        1answer
        122 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
        1answer
        160 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):\...
        1
        vote
        1answer
        239 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 $\|\cdot\|$ ...

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