A structured proof of Kolmogorov's Superposition Theorem

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 $φ_1,\ldots,φ_5 : [\,0, 1\,]\to [\,0,1\,]$ 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 $f(x,y)=\sum\limits_{k=1}^5 h\left(φ_k(x)+\sqrt{2}\,φ_k(y)\right).$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.

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