Suppose $\Sigma \subset \mathbb{R}^d$ and $k \in \mathbb{N}$. I would like to prove that $\Sigma$ is a $k$-smooth manifold if and only if it is locally a graph of an opportune function.
I mean that for every $x \in \Sigma$ there exists an open set $U$ and a function $h : \mathbb{R}^k \to \mathbb{R}^{d-k}$ such that $\Sigma \cap U=graph(h)\cap U$
I tried using a regular parametrization $\varphi$ and using the fact that the rank of $d \varphi$ is $k$ but I do not know where to start.