$\mathbf{C^2}$-Lusin approximation of convex functions: one variable case

Abstract: We prove that if $f:(a,b)\to\mathbb{R}$ is convex, then for any $\varepsilon>0$ there is a convex function $g\in C^2(a,b)$ such that $|{f\neq g}|<\varepsilon$ and $\Vert f-g\Vert_\infty<\varepsilon$.