Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a continuously differentiable mapping. We assume that the set $$\{x\in\mathbb{R}^{n};j(f)(x)=0\}$$ is a hypersurface of $\mathbb{R}^{n}$, where $j(f)(x)$ denotes the jacobian determinant of $f$ at $x$.

I wonder if it is possible to apply a weak variant of the Inverse theorem function to $f$ at any point of $\mathbb{R}^{n}$ which states as follow:

For a given $a\in\mathbb{R}^{n}$, there is some neighborhood $V$ of $a$ in $\mathbb{R}^{n}$ such that $f(V)$ is a neighborhood of $f(a)$.