SCHWEITZER 1999 Problem 7

let $f:R\to R$ be a continuous function tf(t)>0 for $t\neq 0$ . Prove that there exists a non-zero differentiable function $y:[0,\infty)\to R$ such that $y'(t)=f(y(t-1))\,\forall t>1$ and the roots of y are bounded.