SCHWEITZER 1957 Problem 4

Let $F_{\epsilon} (0<\epsilon<1)$ denote the class of non-negative piecewise continuous functions defined on $[0,\infty)$ which satisfy the following condition: $f(x)f(y)\leq \epsilon^{\mid x-y\mid} (x,y \geq 0)$ . Find the value of $s_{\epsilon}= \sup_{f\in F_{\epsilon}} \int_{0}^{\infty} f(x) dx$ (R. 5)