Let $U,V$ open sets in $\Bbb{R}^n$, $f:U\rightarrow V$ a diffeomorphism of class $C^2$. I need to prove that, for all $a\in U$, exists $r>0$ such that the image of the open ball centered at $a$ with radius $\epsilon$ is convex, for all $\epsilon\leq r$.
My idea is very simple, and that is why I think that I'm forgetting something. I found some different answers for that question, like this one. But I want to know what is wrong about my thinking.
Well, in fact I only used the fact that $f^{-1}$ is continuous and $f$ is surjective. If $a\in U$, let $A$ an open ball of $a$. So, $f(A)$ is a open set. So, there exists $r>0$ such that $B_{f(a)}(r)\subset f(A)$. So, I can use these balls as the convex questions.
What is wrong? And what can I do to solve the question? I did not understand the answerd linked above, since it uses some statements about the Hessian, and I didn't study hessians yet. The context is inverse funcion theorem.