SEEMOUS 2008 Problem 1

Let $f:[1,\infty)\to(0,\infty)$ be a continuous function. Assume that for every $a>0$ , the equation $f(x)=ax$ has at least one solution in the interval $[1,\infty)$ . (a) Prove that for every $a>0$ , the equation $f(x)=ax$ has infinitely many solutions. (b) Give an example of a strictly increasing continuous function $f$ with these properties.