 Solution

A function is an analytic function at $z_0$ if it is holomorphic throughout some open neighbourhood of $z_0$, equivalently if it has a convergent power series there. In real components $g=u+iv$, continuous first partial derivatives in a neighbourhood together with the Cauchy-Riemann equationsthroughout that neighbourhood are sufficient, and for an analytic function they hold there. Satisfying these equations only at the single point is not enough to establish analyticity in a neighbourhood.