# 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?

• Have you checked any other functions? – user44191 Feb 21 at 2:29
• see answer below -- it seems any continous function applies this equation – ZUN LI Feb 21 at 17:36

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.