Preimage of 0 for a differentiable function.

Question

If a subset $N$ of a manifold $M$ can be written as $f^{-1}(\{0\})$ being $f:M \longrightarrow \mathbb{R}$ a differentiable function, can I conclude that $N$ is a submanifold of $M$?

Answer 1

Oh no, you can't: every closed subset of $\mathbb R^n$ is the zero set of a $\mathcal C^\infty$ map $f:\mathbb R^n\to \mathbb R$.
Unbelievable? Maybe.
True? Definitely: this a theorem of Whitney, proved in Madsen-Tornehave page 224.

Answer 2

Not necessarily. Consider for example $M=\mathbb R^2$ and $f(x,y)=xy$. Or, for a more dramatic example, $$ f(x,y) = \begin{cases}0 & x\le 0 \\ ye^{-1/x} & x > 0 \end{cases}$$