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D.A. Sprecher showed (https://www.researchgate.net/profile/David_Sprecher2/publication/243052898_A_Representation_Theorem_for_Continuous_Functions_of_Several_Variables/links/554929f20cf2ebfd8e3ad956.pdf) that any continuous function $$f:[0,1]^n \to \mathbb{R}$$ can be represented as

$$f(x_1,...,x_n) = \Phi(\sum_{i=1}^n h_i(x_i))$$

and states that ``clearly the function $\Phi$ must be discontinuous''.

My question is, why must $\Phi$ be discontinuous?

Sprecher mentions that V.I. Arnold (in a Russian paper) showed that even $f(x,y) = xy$ cannot be written as $\Phi(h_1(x) + h_2(y))$ but this is not clear to me since (if we are in a domain contained in $\{(x,y) : x >0,y>0\}$) we can write $xy = e^{\log(x) + \log(y)}$.

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    $\begingroup$ First, Spretcher says "Clearly, the function $\Phi$ will, in general, be discontinuous", which is not what you quote. $\endgroup$
    – VictorZurkowski
    Commented Aug 22, 2016 at 7:22
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    $\begingroup$ Second, Spretcher result is that $h_1, .., h_n$ exists with the universal property that any continuous f can be represented as a function of $\sum_1^n h_i$, the sum of the particular functions $h_1, .., h_n$ whose existance Spretcher show. $\endgroup$
    – VictorZurkowski
    Commented Aug 22, 2016 at 7:27
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    $\begingroup$ Third, the result of Arnold is that $f(x,y)=xy$, with $(x,y) \in [0,1]^2$ cannot be represented as $\Phi(h_1(x) + h_2(y))$ with $\Phi, h_1, h_2$ continuous. Your function has a different domain, so it is not the same of which Arnold spoke. $\endgroup$
    – VictorZurkowski
    Commented Aug 22, 2016 at 7:33
  • $\begingroup$ Thanks for clarifying the results. Would you know the outline of Arnold's proof? $\endgroup$
    – Asterix
    Commented Aug 22, 2016 at 13:09
  • $\begingroup$ No worries. Thanks for the answer! $\endgroup$
    – Asterix
    Commented Aug 22, 2016 at 19:27

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