Is there a continuous and satisfying a certain condition but not integrable function?

Question

Is there a continuous function $f$ on the interval $[0,1]$ which satisfies $$ |f(x+h)+f(x-h)-2f(x)|\leq \mathrm{const} \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, \frac{1}{2}] $$ for all $x,h\in [0,1]$ such that $x-h, x+h \in [0,1]$ and has bounded of variation but $f'$ is not integrable on $[0,1]$?