Let $X,Y$ be normed spaces. A function on open set $U \subset X$ is differentiable at $a\in A$ if there is a linear map $L_a$ tangent to it, and since $a\mapsto L_a$ is again a function on $U$, inductively we can define infinitely differentiable function. I want to extend this definition to arbitary subset $A\subset X$. But the problem is such defined differential might be not unique. For example, $A$ might be a set lying in a proper linear subspace of $X$. So my first question is, is there a natural class of subsets of $X$ such that we can well defined infinitely differentiable on it? For example, for $X=\mathbb R^n$, the class of n dimensional submanifolds of $X$ with boundary?
When this is done, I want to investigate how this definition is related to smoothness:
$f:A\to Y$ smooth at $a$ if there is a smooth extension of $f$ on some neighborhood of $a$.
Does Infinite differentiability implies smoothness?
In particular, for upperhalf plane $H\subset \mathbb R^n$, is a infinitely differentiable function $f:H\to \mathbb R$ neccesarily smooth? (In dimension one, I think $-f(-x)$ might give a smooth extension?)