I have tried solving this problem with this strategy. We know that if a function has a power series representation with radius of convergence R, it is analytic on that radius. I can clearly get infinite radius of convergence for the upper part of the piecewise function, implying that g is analytic everywhere except maybe 0. I need to check that g(z) is differentiable at 0, but unfortunately, we are not allowed to use L'Hoptial's rule. I know the limit of the upper half as g approaches 0 is 1/2, by looking at the series expansion. Any ides on how to proceed?
As you noted, the only possible issue is at $0$. The upper piece has a singularity at $0$; try determining what kind of singularity it is.
Since the power series converges at 0 to g(0), we know more than that g is continuous at 0 -- we know g is analytic at 0.