Independence of Smooth Extension on Jacobian at a Boundary Point

Question

Say $A \in \mathbb{R}^n$ is any subset, not necessarily open, and $F : A \to \mathbb{R}^m$. Smoothness is initially defined only on open subsets, but we traditionally say that $F$ is smooth on $A$ if it extends to a smooth function in a neighborhood of each point. If $F$ is smooth, then does the Jacobian we compute at a boundary point of $A$ depend on the smooth extension we choose? Certainly it would be problematic if it did, but I'm having trouble proving this for myself.