# Tagged Questions

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### 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 ...
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### Stromquist's 3 knives procedure

(copied from math.SE) BACKGROUND: A cake has to be divided among 3 people with possibly different tastes, such that each person receives a single connected piece, and no person prefers another person'...
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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 ...
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### Lipschitz continuity of singular values

How smooth are the singular values of a matrix F in terms of entries of F? I am hoping for Lipschitz continuity, but was not able to find it.
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### Cardinality of $C^*([0,1])$ [closed]

What is the cardinality of the continuous dual of $C([0,1])$ (the set of continuous functions from $[0,1]\to \mathbb{R}$)?
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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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### 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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### Is a continuous map between smoothable manifolds of the same dimension always smoothable?

(My question is inspired by this math.SE question, whose negative answer I showed by a dimension-increasing map.) Is it the case that for all smoothable manifolds $M_0$ and $M_1$ with the same ...
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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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### 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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### Find a continuous function with a prescribed continuity set

It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$. In the book "Understanding Analysis" by Abbott is stated in page 128 ...
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### Is the Poincare action on the Klein-Gordon quantum field strongly continuous?

I am interested in checking continuity property of the Poincare group action on the Klein-Gordon quantum field theory defined over the Minkowski spacetime. Maybe the simplest example of QFT out there. ...
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### Absolute continuity and the Luzin N-Property for functions of two variables

It is a well known fact that absolutely continuous functions of one real variable have the so-called Luzin N-property. That is, if $E\subset\operatorname{Domain}(f)$ has zero measure, then $f(E)$ has ...
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### 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 ...
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### Uniform approximation of increasing function in $C^{\infty}$

I have an increasing continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$ which is not differentiable everywhere, and I would like to approximate it with an infinitely differentiable function \$g\...