Consider $\textbf{f}: U\subset \mathbb{R}^n\to \mathbb{R}^n$ and $\textbf{a} \in U$. Suppose that $\textbf{a}$ is an interior point of $U$ and $\textbf{f}$ is differentiable at $\textbf{a}$ with $\det(D_\mathbf{a} \mathbf{f})\ne 0$.
The question asks whether $\textbf{f}(\textbf{a})$ is an interior point of $\textbf{f}(U)$. I am thinking it is going to use the fact that $\textbf{f}$ is invertible on some open set around $\textbf{a}$ and perhaps do an intersection of this set with $B(\textbf{a},\epsilon)$, a subset of $U$. But I don't know how to proceed.
Any help will be appreciated.
$\mathrm{det}(d_af)\neq 0$ so as you suggested, by the inverse function theorem there is a open subset $U$ containing $a$, and an open subset $V$ containing $f(a)$, such that the induced map $f:U\to V$ is a diffeomorphism. Then, $V$ is an open neighborhood of $f(a)$, which is contained in $\mathrm{Im}f$, so that $f(a)$ is an interior point of $\mathrm{Im}f$.