Additively stable sets, critical sets for the $3k-4$ theorem in $\mathbb Z$ and $\mathbb R$

We describe additively left-stable sets, i.e. sets satisfying $((A+A)-\inf (A))\cap [\inf (A),\sup (A)]=A$ (meaning that $A-\inf (A)$ is stable by addition with itself on its convex hull), when $A$ is a finite set of integers and when $A$ is a bounded set