We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilbert's 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are continuous functions $\varphi_1,\ldots,\varphi_5 : [\,0, 1\,]\to [\,0,1\,]$ such that for any continuous function $f: [\,0,1\,]^2\to\mathbb R$ there is a continuous function $h: [\,0,3\,]\to\mathbb R$ such that for any $x,y\in [\,0, 1\,]$ we have $$f(x,y)=\sum\limits_{k=1}^5 h\left(\varphi_k(x)+\sqrt{2}\,\varphi_k(y)\right).$$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.
翻译:我们展示了一个非常有条理的详细解释, 证明Hilbert第13个问题有以下值得庆祝的结果。 对于两个变量的函数, 声明如下。 Kolmogorov Theorem。 有连续的函数 $\ varphi_ 1,\ ldots,\ varphi_ 5 : [\, 0, 1\,\,\\,]\ to\mathb R$ : [\, 0, 1\\,] \\\ t\mathb R$, 这样对于任何连续的函数, $f: [\, 0, 3\,\]\ to\mathbb R$ 具有持续性的函数 : [\, 0, 3\]\\\\\\\\\\ mathb R$, 对于任何只熟悉连续功能基本特性的学生来说, 。 证明是非专业专家, 特别是熟悉连续函数的学生可以获取的 $x,y\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\