Let $f\colon \mathbb R^n \times \mathbb R^m \to \mathbb R^m$, $\mathbb R^n \times \mathbb R^m \ni (x, y) \mapsto f(x,y)$ be continuously differentiable. The implicit function theorem guarantees, under certain conditions, that if there exists a point $(a, b) \in \mathbb R^n \times \mathbb R^m$ such that $f(a, b) = 0$, then in a neighborhood $U \subset \mathbb R^n$ of $a$ there exists a $g \colon U \to \mathbb R^m$ which is continuously differentiable and satisfying $f(x,g(x)) =0$. In practice, this guarantees that we can write $y = g(x)$, at least locally.
I am wondering whether it is possible to obtain a similar result for a function $f\colon \mathbb R^n \times \mathbb R^m \to \mathbb R$. If I find a point $(a,b)$ as above such that $f(a, b) = 0$, is it possible, at least in some neighborhood of $a$, to write $y = g(x)$? If yes, under which conditions?