Let $A \subset \mathbb{R}^n$ be open, let $f: \mathbb{R}^n$ be continuously differentiable, and suppose that $det(D_xf) \neq 0$ for all $x$ Show that $f(A)$ is an open subset of $\mathbb{R}^n$.
My Attempt:
Given that the determinant of the differential for all $x$ does not equal $0$ and that $f$ is continuously differentiable (class $C^ \infty),$ we can use the inverse function theorem for all $x\in A$.
Now we know that there exists open sets $U,V$ in $\mathbb{R}^n$ and a function $f:V \rightarrow \mathbb{R}^n$ with some conditions.
How do I prove this using the inverse function theorem?