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There are no continuous solutions. Since the cosine has a unique fixed point $x_0$ (such that $\cos x_0=x_0$), it should be a fixed point of $f$. And f should be injective and hence monotone (increasi …

No, such functions do not exist. More precisely, let $f:\mathbb R\to\mathbb R$ be an arbitrary function, $\Sigma$ is the set of $x\in\mathbb R$ such that $f'(x)$ exists and equals 0. Then $f(\Sigma)$ …

I like a recent proof by Akopyan and Tarasov: A. V. Akopyan, A. S. Tarasov, "A constructive proof of Kirszbraun's theorem"(Russian), Mat. Zametki 84 (2008), no. 5, 781--784; translation in Math. Note …

I found that I need to use the following facts in a paper that I am writing. Let $f\in C^\infty(\mathbb R)$, then If $f(0)=0$, then $f(x)=x g(x)$ for some $g\in C^\infty(\mathbb R)$. If $f$ is even …

The statement is true. It is almost precisely Lemma 2 in the paper D.Burago, "Periodic metrics", Adv. Soviet Math. 9, (1992), 205-210. The proof is short but not easy to invent. The paper can be read …

You can define $A(x_1,x_2,\dots,x_n)=B(x_1,x_2)$ where $B$ is a solution for $n=2$. As for $B$, one can e.g. define it on the unit circle by $B(\sin t,\cos t)=\cos 3t$ and extend by homogeneity.

Yes. Incidentally, just recently I had to write down a proof of a similar fact in one of my papers. It is quite technical. Let us work at the endpoint $x=0$. We have to prove that the function $x\map …

No. Consider $f(x,y)=e^x+y^2$, then $\varphi(x,y)=(e^x,2y)/(e^x+y^2)$. The image of $\varphi$ has only one point $(1,0)$ on the axis $y=0$. The points $a:=\varphi(0,1)=(\frac12,1)$ and $b:=\varphi(0,- …

Here is a simple proof that the property holds only for Euclidean norms, at least if the norm in question is $C^1$ smooth and strictly convex. Surely it was known way before Gromov was born. Let $S$ …

It seems that the following interpretation is not covered yet: the domain is allowed to change but $f$ is not required to be proper. In this case the answer is no. Take any connected domain on the pla …

You are essentially asking whether a non-expanding linear map $A:(\pi,\|\cdot\|_\infty)\to(\mathbb R^3,\|\cdot\|_\infty)$ can be extended to a non-expanding linear map $B:(\mathbb R^3,\|\cdot\|_\infty …

This is a follow-up to this question (in fact, this is what originally motivated me to ask that one.) Let's say that a sequence $\{s_i\}$ of positive reals covers a set $X\subset\mathbb R$ if there i …

Here is a formalization of André Henriques' answer to the Hausdorff dimension variant of the question. Let $K=\{0,1\}^\infty$ be the standard Cantor set. Define a map $f:K\to[0,1]$ as follows: for a …

The following extension of Keivan Karai's comment proves the result. Consider $q(x):=p(x)-c$ where $c=\frac12 \min p([0,1])$. Approximate $q(x)$ as Keivan suggested: let $$ q_n(x) = \sum_{i=0}^n q( …

The sum $\sum a_n/n$ can be negative. Below I construct a finite sequence; one can always add a negligibly small tail to get infinitely many non-zeroes. Begin with $a_1=1$ and $a_2=-1$. This gives $A …