I remenber here the concept of a contraction mapping.
Definition: Let (X, d) be a metric space. Then a map T : X → X is called a contraction mapping on X if there exists q ∈ [0, 1) such that
$$ d(T(x),T(y))\le qd(x,y) d(T(x),T(y))\le q d(x,y) \ \ x, y \in X.$$
Assume that, $f :B \longrightarrow f(B) \subset B$ is a bijective contraction mapping and $B$ is a closed ball. I'd like to know if always $$ f(\partial B)=\partial (f(B)) ? $$
I have a feeling that this statement is true.