Implicitly defined function

Let $F$ be a function on a subset of $\mathbb R^{n+m}$ . A function $\varphi:U\to\mathbb R^m$ , where $U\subseteq\mathbb R^n$ , is implicitly defined by

F(x,y)=0

on $U$ if $F(x,\varphi(x))=0$ for every $x\in U$ .

Remarks

An equation need not determine $y$ uniquely or even determine it at all. The implicit function theorem gives local existence and uniqueness near a solution $(x_0,y_0)$ when $F$ is continuously differentiable and the derivative with respect to $y$ is invertible there.

Examples

The equation $x^2+y^2-1=0$ implicitly defines $y=\sqrt{1-x^2}$ near $(0,1)$ , and $y=-\sqrt{1-x^2}$ near $(0,-1)$ .
The equation $x+y+z=0$ implicitly defines $z=-(x+y)$ as a function of $(x,y)$ on all of $\mathbb R^2$ .