[math-fun] Riemann integration question

I've got another question arising from the honors calculus course I teach. Suppose f is continuous on [a,b] with I = int_a^b f(x) dx, and for every epsilon > 0 let delta(epsilon) be the largest delta > 0 such that every Riemann sum arising from a partition of [a,b] with mesh less than delta differs from I by less than epsilon. Is it true that (leaving aside the case where f is constant) delta(epsilon) goes to zero like epsilon^2, in the sense that delta(epsilon)/epsilon^2 is bounded above and below by constants as epsilon goes to zero? Jim Propp