Leave $ U, V $ open sets in $ Bbb {R} ^ n $, $ f: U rightarrow V $ a class dipheomorphism $ C ^ 2 $. I need to demonstrate that, for everyone $ a in U $exists $ r> 0 $ such that the image of the open ball centered on $ to $ with radio $ epsilon $ it's convex, for everyone $ epsilon leq r $.

My idea is very simple, and that's why I think I'm forgetting something. I found some different answers for that question, like this one. But I want to know what is wrong with my thinking.

Well i actually just used the fact that $ f ^ – $ is continuous and $ f $ it is surjective. Yes $ a in U $, leave $ A $ an open ball of $ to $. So, $ f (A) $ It is an open set. So there is $ r> 0 $ such that $ B_ {f (a)} (r) subset f (A) $. So, I can use these balls as convex questions.

It's wrong? And what can I do to solve the question? I did not understand the answer linked above as it uses some statements about the Hessian, and I have not studied hessians yet. Context is the inverse function theorem.