Proving this subset of $\mathbb{R}^{2}$ is not a submanifold of $\mathbb{R}^{2}$

Question

I am trying to show that the set $$X = \{(x, y) \in \mathbb{R}^{2} \mid x^{2}=y^{3}\} \subset \mathbb{R}^{2} $$ is not a submanifold of $\mathbb{R}^{2}$. I understand that the curve isn't smooth at $(0,0)$ and this is why, but I just can't translate this into a proper proof. I know it has to do with the implicit function theorem (or inverse function theorem?) but I'm not sure how those theorems help here. The explanations for similar examples that I have seen make use of these tools but they are a little hand-wavey and I just don't see how to formalise the argument.

Answer 1

Let suppose that $X$ is a manifold, then locally in $(0,0)$ the set $X$ is the graph of a differentiable function $g$, which satisfies $g(0)=0$. Then $x^3=g(x)^2$. If $n$ is the order of $g$ at $0$ then $2n=3$. Contradiction.