Let $f:A\rightarrow\mathbb{R}^k$ for some $A\subset\mathbb{R}^n$ not open, for example we could have $A=\{(x,y)\in\mathbb{R}^2:y=x^2\}$ which is a "nice set" by all accounts. If for every $x,y,z\in A$ sufficiently close to each other, and every analytic path $a(t):[-1,1]\rightarrow A$ satisfying $a(-1)=x$, $a(0)=y$, $a(1)=z$, the corresponding $(a(t),f(a(t))$ is an analytic path in $\mathbb{R}^{n+k}$, can we say that $f$ is analytic in any meaningful way? Does it help if $A$ is a submanifold of $\mathbb{R}^n$?
I'm looking more for confirmation that this concept exists in literature and has been codified formally, I'm not asking for any sort of help musing.