While reading through a paper I came a cross a result (due to the Implicit Function Theorem) that I cannot derive.
Let $u(x,y)$ be the solution of a PDE ($x$ and $y$ are independent variables). We can expand the solution around an arbitrary non-characteristic singularity manifold given by $g(x,y)=0$ in a power series of the form $\displaystyle\sum_{n=0}^{\infty}a_{n}(x,y)g(x,y)^{n+\alpha},$ where $\alpha$ is a negative integer (to be found using leading order analysis).
The following result is the one that I cannot derive.
Given that $g=0$ is assumed to be non-characteristic we have $g_{y}\neq 0$. Thus, by the Implicit Function Theorem we have $g(x,y)=y-f(x)$ near $g=0$, where $f(x)$ is an arbitrary function of $x$.
How do I get $g(x,y)=y-f(x)$ near $g=0$?