Let $(M,J)$ be an almost complex manifold and $N\subset M$ a closed almost complex submanifold, i.e. the tangent space $T_xN$ at every point $x\in N$ is invariant under $J$.
Let $\Sigma$ be a connected Riemann surface. Since the question I am going to ask is local in nature, we may assume $\Sigma$ to be a disk. Let $u:\Sigma\rightarrow M$ be a $J$-holomorphic curve.
Assume there is a non-empty open subset $V\subset \Sigma$ such that $u(V)\subset N$. Does it follow that the image $u(\Sigma)$ of $u$ lies entirely in $N$?
Two remarks:
(1) It is proved in e.g. McDuff-Salamon's book that this is true if $N$ is a point.
(2) This question is obvious if $J$ is integrable because $N$ then admits local defining functions so that we can apply (1).
But for general almost complex submanifold $N$, I don't know if local defining functions exist or not.
Edit: By local defining functions I mean J-holomorphic functions on neighbourhoods of points of N whose zero locus is N.