= 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 equations>
$$
\boxed{u_x=v_y,\qquad u_y=-v_x}
$$
throughout 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.
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