A problem on continuous and periodic functions (Q1057895)

A function f on the real line is said to have period 1 if \(f(x+1)=f(x)\) for all x. Problem (Chung-Erdős): If f is continuous and of period 1, and \(d_ j\) (1\(\leq j\leq n)\) are given, is there a rational r such that \(f(r)\leq f(r+d_ j)\) for all j ? This paper proves the following theorem: Assume either that every \(d_ j-d_ 1\) is rational, or that the set of minimum and maximum points of f on the open interval (0,1) is finite. Then there are rationals r and r' such that \(f(r)\leq f(r+d_ j)\) and \(f(r')\geq f(r'+d_ j)\) for all j.