For $n \geq 2$ we say a continuous function $f: \mathbb R^n \to \mathbb R^n$ such that the image of any bounded open ball is a bounded open ball of different radius is a balloon function.

Compositions of non-trivial scalings, rotations and reflections can be seen to be balloon functions. Are there any others besides these?

For $n \geq 2$ we say a continuous function $f: \mathbb R^n \to \mathbb R^n$ such that the image of any bounded open ball is a bounded open ball of different radius is a balloon function.

Compositions of non-trivial scalings, rotations, translations and reflections can be seen to be balloon functions. Are there any others besides these?

For $n \geq 2$ we say a continuous function $f: \mathbb R^n \to \mathbb R^n$ such that the image of any bounded open ball is a bounded open ball of different radius is a balloon function.

Are scalings by a non-zero scalar centered at a point the only balloon functions?

For $n \geq 2$ we say a continuous function $f: \mathbb R^n \to \mathbb R^n$ such that the image of any bounded open ball is a bounded open ball of different radius is a balloon function.

Compositions of non-trivial scalings, rotations and reflections can be seen to be balloon functions. Are there any others besides these?