I am wondering how does the smoothness condition of a potential function imply the smoothness of its equipotential line. I would like to know if there is some theorem on this kind of questions.
The smoothness condition could be anything like continuous, Lipschitz continuous, differentiable, analytical, etc.
For example, let $f:\mathbb{R}^2\to \mathbb{R}$ be a differentiable potential function, assume we have a parametrized curve $c(t)$ where $f(c(t))=0$ for all $t\in[0,1]$, what can we say about this curve $c$? Is it, for example, differentiable?
Any help would be appreciated!
The Implicit Function Theorem is what you need. If $f$ has continuous partial derivatives and $\nabla f(c(t_0))\ne(0,0)$ for some $t_0$, then $c$ is continuously differentiable on a neighborhood of $c(t_0)$.