Open set under a function $f$

Question

Let $A$ be open in $\mathbb{R^n}$; let $f :A \rightarrow \mathbb{R^n}$ be of class $C^r;$ assume $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not one-to-one on $A$, the set $B = f(A)$ is open in $\mathbb{R^n}.$

Thanks for any hint!