Is there an elementary example of a function f, such that $|f(x+t)+f(x-t)-2f(x)|/|t|^a\le C$, where $a>1$, such that $f$ is not $C^1$?



Second difference
Yes, i think. 


Pick an integer $n>1$ and let $f$ be the function given by $f(x) = x^n\sin(x^{-n})$ for $x>0$ and $f(x)=0$ for $x\leq 0$. It is continuous, and smooth except at $x=0$. It is not $C^1$ at zero, because the derivative for $x>0$ is $nx^{n-1}\sin(x^{-n}) - nx^{-1}\cos(x^{-n})$, which enters in a state of flailing tantrum close to $0$.


On the other hand, we have $|f(0+t)+f(0-t)-2f(0)| = |f(t)|$, so every $\alpha < n$ is good.


