 Solution

Let $A$ be the set of points with a neighbourhood on which $f=g$. It is nonempty and open. If $p$ is in its closure, continuity and Hausdorffness give $f(p)=g(p)$. Choose a common target chart and a source disc around $p$ on which both images lie in that chart. Their coordinate difference is holomorphic and vanishes at accumulating points from $A$, so the identity theorem makes it vanish on the disc. Thus $p\in A$. Therefore $A$ is also closed, and connectedness implies $f=g$ on all of $R$.