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? 



A functional equation in real analysis
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.](https://en.wikipedia.org/wiki/Bernstein_polynomial#Approximating_continuous_functions) 


