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?