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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.

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    $\begingroup$ Sketch of a possible proof: Write $f(x)$ as the sum of a quadratic function $g(x)$ and a function $h(x)$ satisfying $h(0)=h(1)$ and $h'(0)=h'(1)$. Prove the $o(1/n)$ estimate for $g(x)$ by direct computation and for $h(x)$ by Fourier methods (treating $h$ as a differentiable function on ${\bf R}/{\bf Z}$) and then add to get the $o(1/n)$ estimate for $f$. But is the needed lemma about differentiable functions on the circle true? $\endgroup$
    – James Propp
    Commented Feb 19, 2014 at 17:05

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