Let $f:\mathbb{R}^2 \to \mathbb{R}, \ f(x,y) = x^3+y^3-3xy$, then $M = \{(x,y) \in \mathbb{R}^2 : f(x,y) = 0 \}$ is not a $1$-dimensional submanifold of $\mathbb{R}^1$.
I have two questions.
1) The given solution merely states that $M$ is defined as set of roots of a $C^{\infty}$-function, so it suffices to check that $f$ does not have full rank in $(0,0)$.
However, I only know that characterization that "some $M$ is a manifold if and only if there exists such $f$ ...". So is it possibly true that a stronger result holds? Something in the spirit of
"Let $M$ be locally defined as the roots of a $C^1$ function. If it does not have full rank in $p \in M$, then $M$ is not a manifold."?
2) How would you show it otherwise? I'd reckon you could argue that the number of connected components doesn't match with those of some interval in $\mathbb{R}^1$. Though, I'm not too sure if it's easy to prove that $M$ looks like that.