Q:

A functional equation in real analysis

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:

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. 

