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For what function $u:[0,1]\rightarrow R$ with bounded derivative, such that $\forall p\in[0,1]$,

$\lim\limits_{n\rightarrow\infty}\sum\limits_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k}u(\frac{k}{n})=u(p)$

Could it more than linear functions?

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  • $\begingroup$ Have you checked any other functions? $\endgroup$ – user44191 Feb 21 at 2:29
  • $\begingroup$ see answer below -- it seems any continous function applies this equation $\endgroup$ – ZUN LI Feb 21 at 17:36
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A theorem of Sergei Bernstein says that if $u$ is continuous, then the sequence of functions on the left-hand side converges uniformly to $u$ on $[0,1]$. The polynomials on the left hand side are called the Bernstein polynomials.

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