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I have proved a few years ago the following proposition: There exists $f: [0,1] \to [0,1]$ with Darboux property such that there exist $A,B \subset[0,1]$ with $A\cap B=\emptyset,\ A \cup B=[0,1]$ …

Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros $\alpha_1,...,\alpha_n$. Define the following 'distance': $$ d(\alpha_i,\alpha_j)=\inf\{\int_0^1 \sqrt …

A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \in [ …

In Antoine Henrot Michel Pierre - Variation et optimisation de formes, Une analyse geometrique, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of …

Let $\Omega \subset \mathbb{R}^N$ be an open, connected set with Lipschitz boundary, $N \geqslant 2$ and $$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} …

Here is a more problem solving approach, since this problem comes from AoPS. Instead of working with $f$, work with $g = f''$ which is a convex function. Note that it is enough to assume $g(1)=0$. In …