Orientation preserving mappings

Question

I came across the following question: Let $Y=\{(r,\theta,\phi)\in \mathbb R^3:0\leq r<\infty , 0\leq\theta<2\pi, 0\leq \phi\leq\pi \}$ and let $f:Y\rightarrow \mathbb R^3$ such that $f(r,\theta,\phi)=(r\cos\theta\sin\phi,r\sin\theta\sin\phi,r\cos\phi)$. Show that $f$ preserves orientation. My attempt: the jacobian of $f$ is just $r^2\sin\phi$ and we need to show that it is greater then $0$ for all $r$ and $\phi$. It is obvious that for $r>0$ and $\phi\not\in \{0,\pi\}$ we will get that $f$ preserves orientation since $r^2\sin\phi>0$, but what about the boundary points (i.e. $r=0$ or $\phi\in\{0,\pi\}$) ?