I am currently reading Range's Holomorphic functions and integral representations in several complex variables and would like some clarification on the following definition.
Let $M$ be a complex submanifold of $\mathbb{C}^n$ and let $p \in M$. Then a function $f : M \to \mathbb{C}$ is holomorphic at $p$ if $f \circ H$ is holomorphic at $H^{-1}(p)$ for a local parametrisation $H$ of $M$ at $p$.
There is a theorem in the book that says that a subset $M$ of $\mathbb{C}^n$ is a complex submanifold of dimension $k$ if and only if for every $p \in M$, there exist an open neighbourhood $U$ of $p$ in $\mathbb{C}^n$, an open ball $B^{(k)}(a, \varepsilon) \subset \mathbb{C}^k$, and a non-singular holomorphic map $H : B^{(k)}(a, \varepsilon) \to \mathbb{C}^n$ such that $H(B^{(k)}(a, \varepsilon)) = M \cap U$. Here, Range says that a map $H$ which satisfies these conditions is called a local parametrisation of $M$ at $p$.
My uncertainty is how this definition of local parametrisation is compatible with the definition of a holomorphic function on a complex submanifold. With the conditions as stated, $H$ being non-singular says that it is only locally injective. But it seems that in the definition of a holomorphic function on a complex submanifold, we are able to take the inverse of $H$ not only on a smaller subset of $M \cap U$, but on all of $M \cap U$? Can this really be done?
Another reason why I am uncertain with these definitions is that it is left as an exercise for the reader to prove that if $H$ is a local parametrisation of $M$ at $p$ with image $M \cap U$, then $H^{-1} : M \cap U \to \mathbb{C}^{\dim M_p}$ is holomorphic. So it does seem like somehow it is possible to obtain not only a non-singular map, but a bijective one with all the properties of a general local parametrisation. Being able to do this is very unclear to me, so I'd greatly appreciate it if I could get some help. Thanks in advance!
