All of the bounds I've seen for the error of the midpoint method of integration are expressed in terms of the second derivative of the function. What bounds are available when the function is not twice-differentiable? In particular, is it the case that for every continuous function $f$ on $[0,1]$, the midpoint method of estimating $\int_0^1 f(x)\: dx$ has error $o(1/n)$? If the answer to this last question is "no" (and I'm pretty sure it is, unless I've misunderstood the example Carnahan gave in response to my earlier question Dependence of error on mesh for Riemann sums), does the answer change to "yes" if we assume that $f$ has bounded variation? That $f$ is monotone?