Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$?

For discussion of a related question (which led to the formulation of this one), see Error of midpoint method for functions that are not twice-differentiable . Linda Brown Westrick's example there shows that mere continuity of $f$ does not suffice.

I'm listing fourier-analysis as a tag on the off-chance that Fourier methods might be applicable.