On points separating polynomials of norm one (Q1907259)

Using a simple constructive technique the authors show that for any two distinct points \(\alpha\) and \(\beta\) lying in a given interval \(I\), there exists a polynomial \(P_n (x)\) in the unit ball of the usual Banach algebra of real-valued continuous functions on \(I\) such that \(P_n (\alpha)\neq P_n (\beta)\). Here \(n\), the degree of the polynomial depends on the distance between \(\alpha\) and \(\beta\).