# Is every implicit function reparametrized? [closed]

Consider a continously differentiable non-constant function $f:\mathbb{R}^2\to\mathbb{R}$. Define $$K=\{x\in\mathbb{R}^2|f(x)=0\}.$$

I wish to know whether there is a continuously differentiable function $g:\mathbb{R}\to\mathbb{R}^2$ such that $$\forall k\in K,\quad\exists u\in\mathbb{R}, \quad g(u)=k.$$ So it is like $K$ can be expressed as an image of a continuosly differentiable function $g$.

If $f$ is linear there of course we can find $g$. It is still true for any function? Any counter example?

I am thinking $f(x,y)=x^3+y^3-xy$. But, how would we prove it mathematically?

• This question is not suitable here. You should ask it at Math.SE, or learn about the implicit function theorem. Commented May 26, 2016 at 11:19
• @BenoîtKloeckner: Thanks. Implicit function is only valid locally, isn't it? Commented May 26, 2016 at 15:46

Basically because the existence of such function $g$ would mean that the zeroes of your function $f$ is a one-dimensional manifold, which is not true in general.
Take for instance $f(x)=\text{dist}^2(x;C)$, where $C$ is the unit disc: $f$ is smooth, and here $K=C$. But you cannot express the unit disc as the image of a one-parameter continuously differentiable function. Eventually you could do it with a pathologic non-differentiable function.
More generally, as soon as $C$ is a closed convex set, the function $$f(x)=\text{dist}^2(x;C)$$ is a continuously differentiable function (having even a Lipschitz continuous gradient), whose zeroes are exactly $C$. So basically proving your statement would prove in particular that you can parametrize any closed convex set with a differentiable function.
• Here I am talking about the distance to a set, defined by $$\text{dist}^2(x;C):= \inf \{ \Vert x - c \Vert^2 \ | \ c \in C \}.$$ Its smoothness, when $C$ is a closed convex set, is a classic result. You can find it for instance in: * Convex Analysis and Nonlinear Optimization: Theory and Examples (Borwein and Lewis), Theorem 9.2.3 * Convex Optimization in Normed Spaces (Peypouquet), Example 3.38 and Proposition 3.39. Commented May 27, 2016 at 14:05