Let $f:\mathbb{R} \to \mathbb{R}$ is a function with these special properties. $f$ is continuous everywhere. $f$ is not smooth (not infinitely differentiable). $f$ is differentiable only finitely many times everywhere. $f$ belongs to $L^p$. For any $k$ belongs to $\mathbb{N}$ let $E$ be the set of all points where $f$ is differentiable exactly $k$ times then $E$ is not dense in any open subset of $\mathbb{R}$.Is such a function feasible? I need suggestions on ways to construct such a special function.
This is the final version. not going to change it further. I apologize for the inconvinience