I wish to show, using elementary techniques, that if $f(x)=-x_1^2-x_2^2-...-x_\theta^2+x_{\theta+1}^2+...+x_n^2$ is a map from $\mathbb R^n$ to $\mathbb R$, then $f^{-1}(0)$ is not a regular submanifold of $\mathbb R^n$.
Toward a solution, I know that $f$ in any neighborhood of 0 does not have constant rank. But it is not enough to give me any clue, since I must prove that no chart $(U,\psi)$ on $\mathbb R^n$ with $0\in U$ satisfies the regular submanifold property. How must I proceed?
Appendix: According to the comments below, I realized that it seems easier to prove that it is not a manifold in its own right. Deleting zero from any neighborhood of zero in $f^{-1}(0)$ turns it into a nonconnected subset so that no neighborhood of zero in $f^{-1}(0)$ is not homeomorphic to any open set of Euclidean space. Is it correct?