I want to prove the following statement.
Let $U\subset\mathbb{R}^2$ be open, $f:U\to \mathbb{R}$ a smooth function, $p\in U$, and $\lambda = f(p)$. If $df_p = (\frac{\partial f}{\partial x} (p),\frac{\partial f}{\partial y}(p))\neq (0,0)$, then there exists a neighborhood of $p$ in the preimage $C = f^{-1}(\lambda)$ that is the trace of a regular plane curve.
It seems using the continuity of derivative $df$, there exists a neighborhood $U$ of $p$ such that any point in $U$ has a nonzero derivative. But the problem is to show it's a trace of a regular plane curve. Any hints or comments will be appreciated.
The above problem is Exercise 3.21 in Differential geometry of curves and surfaces by Tapp