# Tagged Questions

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### 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 ...
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### Absolutely continuous functions

it is well known that if a function $f:[0,T]\to\mathbb{R}$ satisfies the inequality $$\vert f(t)-f(s)\vert\leq \int_s^t{m(r) dr},$$ for $s<t$ and some $m\in L^1([0,T])$ then $f$ is absolutely ...
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### Continuity in Banach space for non-linear maps

I want to find an example of a Banach space $X$ and a continuous map $f:X\rightarrow X$ such that $f$ is not bounded on the unit ball. I do not doubt that such an example exists, but I cannot make it ...
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### If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?

I have stumbled upon the following problem during my research: Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
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### Follow up: Is continuity preserved when taking comma object?

Here, I asked wether taking lax pullback preserves continuity, but got no precise answer. However, I have found this recent article by Riehl and Verity which proves something very similar, but I can'...
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### Lipschitz continuity of solution set mapping of a parametric convex optimization problem

I have a parametric convex optimization problem: \begin{array}{cl} \underset{x}{\text{minimize}} & f\left(x,z\right)\\ \text{subject to} & g\left(x\right)\leq0 \end{array} where $x$ is the ...
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### Extending a $*$-Representation of $({C_{c}}(G,\mathscr{A}),\star,^{*})$ to a $*$-Representation of $\mathscr{A} \rtimes_{\alpha} G$

Let $(\mathscr{A},G,\alpha)$ be a $C^{*}$-dynamical system, and consider the twisted convolution $*$-algebra $({L^{1}}(G,\mathscr{A}),\star,^{*})$ defined by \begin{align*} \forall \phi,\psi \...
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### the convolution of integrable functions is continuous?

The question is simple but I still can't prove it or contradict it. Here it goes: Suppose $f$ and $g$ are defined on the circle (or, equivalently, $2\pi$ periodic functions) and Lebesgue ...
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### Speed of convergence for Weyl's Equidistribution theorem

If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that $$\frac{1}{n}\sum_{k=0}^{n-1}f(ak)\rightarrow \int_0^1 f(x)dx.$$ Can we ...
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### Function spaces over pseudocompact spaces

Let $K$ be a pseudocompact Tychonoff space that is, a Hausdorff $T_{3.5}$ space for which every continuous function $\varphi \colon K\to \mathbb{C}$ is bounded. Let $\beta$ be the Stone-Cech extension ...
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### upper semicontinuity in C(X)-algebras

Dear fellows, I've stuck on a step of proposition 1.2 of Rieffel's article (continuous field of C*-algebras coming from group cocycles and actions, 1989). I think it basically proves that a C(X)-...
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### Are point sets of the same order type connected by continuous (order type)-preserving motion?

Given two general position point sets in $\mathbb{R}^2$ of the same size and order type, is it possible to continuously move the points of one set until they coincide with those of the other set in ...