My question is motivated by the following: if $f_1, f_2: \mathbb{R}^m \rightarrow \mathbb{R}$ are real analytic functions and they agree on an open set $V$ then we actually have $f_1 \equiv f_2$. But how do real analytic functions depend on their values on $V$? More precisely is the following true:

Suppose $f_\lambda:\mathbb{R}^m \rightarrow \mathbb{R}$ is a family of analytic functions indexed by $\lambda \in (0,1) \subset \mathbb{R}$. Let $V \subset \mathbb{R}^m$ be an open set and assume that the map $(\lambda,y) \rightarrow f_\lambda(y)$ is analytic on $(0,1) \times V$. Then my question is: is the map analytic on $(0,1) \times \mathbb{R}^m$?

I think this should be straightforward, but I am unable to come up with a rigorous argument or a counter example.