# An inequality about the second-order difference

Fix a continuously differentiable but nowhere twice differentiable function $$f$$ on $$\mathbb{R}$$ supported on $$[0,1]$$. Is it true that for all $$x\in[0,1]$$ and all $$\delta$$ sufficiently small \begin{align*} & \sup_{0 Here $$C$$ is allowed to depend on $$f$$, and does not depend on $$\delta$$ and $$x$$.

$$\rhs(\de):=\de^{-1}\sup_{0 while for $$\de=\dfrac1{2(2n+1)\pi}$$ and natural $$n\to\infty$$ $$\lhs(\de):=\sup_{0 So, $$\lhs(\de)$$ is not $$O(\rhs(\de))$$.