Jacobian determinant of a diffeomorphism on the unit shpere must equal $1$ or $-1$

Question

Let $B \in \Bbb R^3$ be an open sphere centered at the origin with radius $1$. Let $f:B \rightarrow B$ be a diffeomorphism, and let $J_F (x)$ be the Jacobian determinant of $F$ in $x$. Show that $\exists_{x \in B} J_F (x) \in \{1,-1\}$.

I'm not sure how to approach this question. Any clues?

Answer 1

Hints:

Volume computation, intermediate value theorem.