Function mapping open ball to open set?

Question

So I have this. Function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is $C^1$ function and at $x_0$ differential $DF(x_0)$ is regular linear operator. There exists $r>0$ such that $f(K(x_0,r))$ is open set. Thats what we know.

Now the question is if $f(K(x_0,r))$ must be open for every $r \gt 0$.

I know that continuous function do not necessarily map open sets to open sets so I don't know how to approach this problem. How does being $C^1$ function and having regular differential at some point affect mapping open balls to open sets.

$K(x,r)$ is open ball in $\mathbb{R}^n$

Answer 1

Take $f: \mathbb{R} \to \mathbb{R}$ given by $f(x)=(x-1)^2$, and $x_0=0$. What is $f(K(x_0,2))$?

Answer 2

For example, try the function $f(x) = x^2$ with $x_0 = 1$, and $r > 1$.