Let $\alpha(s) = (x(s),y(s))$ be the arc length parametrization of a plane, smooth, closed, convex curve, of length $L$. Let $J:(0,L)\to\mathbb{R}$ be a smooth and Bounded variation (BV introduced after comment from Dirk of possibility of blow up at 0 or L) function. Let $\Omega$ be the set of all interior points of the region bounded by the curve $\alpha(s)$.

The problem I would need help for, is to determine $f:\Omega \to \mathbb{R}$, such that

1. Given any $s\_0 \in (0,L)$, and let $p\_0 = \alpha(s\_0)$, $${\lim\_{p\to p\_0}}\_{p\in\Omega}f(p) = J(s\_0)$$
2. $f\in C^{\infty}(\Omega \setminus R)$, where $R$ is a set of 1-Hausdorff measure equal to 0.
3. The total variation $V(f,\Omega)$ is minimum possible.

Motivation : Trying to define step functions in $\mathbb{R}^2$

PS : [![enter image description here](https://i.stack.imgur.com/a1HRO.png)](https://i.stack.imgur.com/a1HRO.png) 

^ Picture source : Wikipedia