MathDB

Consider an integer

n \geq 1

a_1,a_2, \ldots , a_n

real numbers in

[-1,1]

satisfying \begin{align*}a_1+a_2+\ldots +a_n=0 \end{align*} and a function

f: [-1,1] \mapsto \mathbb{R}

such \begin{align*} \mid f(x)-f(y) \mid \le \mid x-y \mid \end{align*} for every

x,y \in [-1,1]

. Prove \begin{align*} \left| f(x) - \frac{f(a_1) +f(a_2) + \ldots + f(a_n)}{n} \right| \le 1 \end{align*} for every

x

\in [-1,1]

. For a given sequence

a_1,a_2, \ldots ,a_n

, Find

f

and

x

so hat the equality holds.

Problem Statement