A rough differentiable function

A real-valued continuously differentiable function $f$ on the unit interval is constructed such that

$$\sum_{k=1}^\infty \beta_f (x, 2^{-k} ) = \infty$$

holds for every $x \in [0,1].$ Here $\beta_f (x, 2^{-k} )$ measures the distance of $f$ to the best approximating linear function at scale $2^{-k}$ around $x$.