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

We prove that if $f:(a,b)\to \mathbb{R}$ is convex, then for any $\epsilon >0$ there is a convex function $g\in C^2(a,b)$ such that $\mathrm{\mid }\{f\mathrm{\ne }g\}\mathrm{\mid }<\epsilon$ and $\mathrm{\parallel }f-g{\mathrm{\parallel }}_{\mathrm{\infty }}<\epsilon$.