Potočný's Theorem

Potočný's Theorem addresses polynomial functions where all derivatives are bounded within certain limits. It states that if a polynomial's derivatives of all orders are uniformly bounded over the entire real line, then the polynomial itself must be of degree zero or one—that is, a constant or linear function. This means that higher-degree polynomials cannot have all their derivatives uniformly bounded everywhere, highlighting a fundamental limit on the behavior of polynomial functions and their derivatives across the entire real line.