If $f(z)=g(z)$ on $(0, \infty)$ and f(z) is holomorphic on an open set $U \subset \mathbf{C}$ with $(0, \infty) \subset U$, but we do not have any information about where $g(z)$ is holomorphic, can we still analytically continue $g(z)$ to $U$ ?
Normally we apply analytic continuations to two functions that are known to be holomorphic in open subsets of the plane, but what if we only know that one of them is holomorphic in an open subset. In my example it follows that since $f(z)$ is holomorphic at any positive real number that $g(z)$ is differentiable on the real line, but that's the only conclusion I can draw.
Best Answer
You question is a bit strange (or it is not clear to me what is your question). Strictly speaking, the information you have given about $f$ means that it is analytic on $U$ and it is an extension of $g$ to $U$.
The values of a function at a set of points that accumulate at a finite point $a$ are enough to determine the germ (the series at $a$) of an analytic function.
The positive real axis accumulates at any of its points. If there is an analytic function equal to $g$ there is only one near each point of $(0,\infty)$. We are told $f$ is that function.