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?