Is it possible to represent (up to a $λ^3$-null set) a 2-dimensional submanifold of $R^3$ as the graph of a $C^1$-function $f:U⊆R^2\to R$?

Question

Let $M$ be a two-dimensional submanifold of $\mathbb R^3$. Is $M$ (globally) representable as the graph of a continuous differentiable function $f:U\subseteq\mathbb R^2\to\mathbb R$ in the sense, that $f(U)\cup N=M$ for some $\lambda^3$-null set $N$.

I know that each submanifold is locally representable in such a way and I'm curious whether or not a global representation is possible in this special scenario.

[If not, can we "cut" $M$ into mutually disjoint pieces which can be represented in this way?]

Answer 1

No, just consider the sphere $S^2$ (which is indeed a manifold because $S^2=\{(x,y,z)\in\mathbb{R}^3\,;\,x^2+y^2+z^2-1=0\}$). For your edit, you can show that : $M$ is a submanifold of $\mathbb{R}^n$ of dimension $m$ if for all $a\in M,$ it exists an open neighbourhood $U$ of $a$ in $\mathbb{R}^n,$ an open set $V\subset\mathbb{R}^n$, a map $\Gamma\in C^1(V,\mathbb{R}^{n-m})$ and an isometry $i$ such as : $$U\cap M=i(\{(x,\Gamma(x))\,;\,x\in V\})$$ which is the local definition you are looking for.

Answer 2

It is false : let's consider $$P:=\{(x,0,z)\in\mathbb{R}^3\,;\, x,z\in\mathbb{R}\}.$$ It is a $2-$dimensional submanifold of $\mathbb{R}^3$ as a subvectorial space of dimension $2$. See as a graph, your function would have to take a uncountable number of values and so you can't even do the construction that you want (the $f_n$ would have an uncountable number of values to take too).