Let $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 }$ be a smooth function whose derivative at every point is non-singular. Suppose that $f ( 0 ) = 0$ and for all $v \in \mathbb { R } ^ { 2 }$ with $| v | = 1 , | f ( v ) | \geq 1$. Let $D$ denote the open unit ball $\{ v : | v | < 1 \}$. Show that $D \subset f ( D )$. (Hint: Show that $f ( D ) \cap D$ is closed in $D$.)