Real-analytic maps between open subsets of the coordinate plane

A smooth map $F=(u,v):U\to {\mathbb{R}}^2$ on an open set $U\subseteq {\mathbb{R}}^2$ is real analytic when, at every $a\in U$, both components equal their total-degree Taylor series on some neighbourhood of $a$.

More explicitly, write multi-indices, their factorials, and the derivatives $D^{\alpha }$ as in $C^k$ maps and multi-index derivative notation in Euclidean space and The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$. For each $a\in U$ there must be a neighbourhood on which, for $h=(h_0,h_1)$ with $a+h\in U$,

$u(a+h)={\sum }_{n\ge 0}{\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=n}\frac{D^{\alpha }u(a)}{\alpha !}{h}^{\alpha },v(a+h)={\sum }_{n\ge 0}{\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=n}\frac{D^{\alpha }v(a)}{\alpha !}{h}^{\alpha }\mathrm{.}$

Each inner sum is finite (Finite sums and finite products, by recursion), and each outer sum is a real series in the sense of Series, partial sums, convergence and the sum, divergence, and the tail series. The coordinate functions $u$ and $v$ are the components of the map into ${\mathbb{R}}^2$ under the convention of Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions. Equality with the displayed series includes convergence to the stated component value; convergence is not presumed merely from smoothness.