Consider two functions $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $g: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$. Let $g$ be continuously differentiable with respect to $y$.
I would like to define $f$ as follows: $f(x)=y$, where $y\in\mathbb{R}^m$ is such that $$ \nabla_y g(x,y) = \frac{\partial g}{\partial y}(x,y) = 0. $$ That is, for every $x\in\mathbb{R}^n$, the function $f(x)$ returns a $y\in\mathbb{R}^m$ that solves the equation $\nabla_y g(x,y) = 0$.
What could be a short notation for defining this function?