[proofplan]
The proof reduces the assertion to the identity theorem for real-analytic functions on connected real-analytic manifolds. The analytic orbit theorem identifies the connected CR orbit component $C$ as an immersed real-analytic integral manifold whose inclusion into $M$ is real-analytic. Since $f$ is real-analytic on $M$, its restriction to $C$ is real-analytic; because it vanishes on a non-empty open subset of $C$, the real-analytic identity theorem forces it to vanish on all of $C$.
[/proofplan]
[step:Realize the CR orbit component as a connected real-analytic integral manifold]
For each $q\in M$, let $T_qM$ denote the tangent space of $M$ at $q$, and let $T_q\mathbb C^N$ denote the tangent space of the ambient real manifold underlying $\mathbb C^N$ at $q$. Let
\begin{align*}
TM:=\bigsqcup_{q\in M}T_qM
\end{align*}
denote the tangent bundle of $M$. If $U\subset M$ is open, let
\begin{align*}
TU:=\bigsqcup_{q\in U}T_qM
\end{align*}
denote the restricted tangent bundle over $U$. Let $J:T\mathbb C^N\to T\mathbb C^N$ denote the standard complex structure on the ambient tangent bundle. Define the real CR tangent distribution $H(M)\subset TM$ by
\begin{align*}
H_q(M):=T_qM\cap J(T_qM)
\end{align*}
for $q\in M$. Let $\mathcal X_H$ denote the family of all real-analytic vector fields $X:U\to TU$ defined on open subsets $U\subset M$ such that $X(q)\in H_q(M)$ for every $q\in U$.
By the Nagano-Sussmann orbit theorem for real-analytic vector fields, applied to the family $\mathcal X_H$ on the real-analytic manifold $M$, each [connected component](/page/Connected%20Component) of the orbit of $\mathcal X_H$ is a connected immersed real-analytic submanifold of $M$, and its tangent space at every point is generated by the values of iterated Lie brackets of vector fields in $\mathcal X_H$. Since the CR orbit $\mathcal O(p)$ is, by definition, the orbit generated by the real and imaginary parts of local real-analytic CR vector fields, the given connected component $C$ carries a connected immersed real-analytic manifold structure for which the inclusion map
\begin{align*}
\iota_C:C\to M
\end{align*}
is real-analytic.
Here the invoked analytic orbit theorem is a standard prerequisite not yet linked in the wiki: Nagano-Sussmann orbit theorem for real-analytic vector fields.
[/step]
[step:Restrict the real-analytic CR function to the analytic orbit component]
Define
\begin{align*}
g:C&\to\mathbb C
\end{align*}
\begin{align*}
q&\mapsto f(\iota_C(q)).
\end{align*}
Thus $g=f\circ\iota_C$. Since $f:M\to\mathbb C$ is real-analytic by hypothesis and $\iota_C:C\to M$ is real-analytic by the preceding step, the composition $g:C\to\mathbb C$ is real-analytic.
Moreover, the hypothesis $f|_A=0$ says precisely that for every $q\in A$,
\begin{align*}
g(q)=f(\iota_C(q))=f(q)=0,
\end{align*}
where the last equality uses that $\iota_C$ is the inclusion map. Hence $g$ vanishes on the non-empty open subset $A$ of the connected real-analytic manifold $C$.
[guided]
We now isolate the analytic object to which the identity theorem will be applied. The orbit component $C$ is not merely a subset of $M$; by the analytic orbit theorem it has its own immersed real-analytic manifold structure, and the inclusion
\begin{align*}
\iota_C:C\to M
\end{align*}
is a real-analytic map.
Define the restricted function
\begin{align*}
g:C&\to\mathbb C
\end{align*}
\begin{align*}
q&\mapsto f(\iota_C(q)).
\end{align*}
This is the correct way to regard $f$ on the immersed orbit: it is the composition $f\circ\iota_C$. Since $f:M\to\mathbb C$ is real-analytic and $\iota_C:C\to M$ is real-analytic, the composition theorem for real-analytic maps gives that $g:C\to\mathbb C$ is real-analytic.
The assumption $f|_A=0$ transfers directly to $g$. Indeed, for every $q\in A\subset C$, the inclusion map satisfies $\iota_C(q)=q$ as a point of $M$, so
\begin{align*}
g(q)=f(\iota_C(q))=f(q)=0.
\end{align*}
Thus $g$ is a real-[analytic function](/page/Analytic%20Function) on the connected real-analytic manifold $C$ and vanishes on the non-empty open subset $A\subset C$. This is exactly the situation in which the real-analytic identity theorem applies.
[/guided]
[/step]
[step:Apply the real-analytic identity theorem on the connected orbit component]
By the identity theorem for real-analytic functions on connected real-analytic manifolds, a real-analytic function on a connected real-analytic manifold that vanishes on a non-empty open subset vanishes everywhere on that manifold. Applying this theorem to $g:C\to\mathbb C$, using that $C$ is connected and that $g$ vanishes on the non-empty open subset $A\subset C$, gives
\begin{align*}
g(q)=0
\end{align*}
for every $q\in C$.
Here the invoked identity theorem is a standard prerequisite not yet linked in the wiki: identity theorem for real-analytic functions on connected real-analytic manifolds.
[/step]
[step:Translate the conclusion back to the original CR function]
For every $q\in C$, the equality $g(q)=0$ and the definition $g=f\circ\iota_C$ give
\begin{align*}
f(q)=f(\iota_C(q))=g(q)=0.
\end{align*}
Therefore $f|_C=0$. Since $C$ was the connected component of the CR orbit containing the given non-empty open zero set, this proves the claimed unique continuation along the CR orbit component.
[/step]
