Every integer occurs as the degree of a sphere map

For $d=0$ take a constant map, which has degree zero by F2.

For $k=\mathrm{\mid }d\mathrm{\mid }>0$, choose $k$ disjoint closed coordinate balls with nonempty interiors. Collapse their boundaries and the complement of their interiors to a single point. The quotient is a wedge of $k$ copies of $D^n\mathrm{/}\mathrm{\partial }{D}^n\cong S^n$: a homeomorphism of the ball interior with ${\mathbb{R}}^n$, followed by inverse stereographic projection, extends to this quotient by sending the boundary to the omitted pole. Fold these copies to a common target sphere. Use an orientation-preserving homeomorphism on each copy if $d>0$, and compose each with a coordinate reflection if $d<0$. The maps agree at the collapsed point, so the quotient construction is continuous.

A point distinct from the common pole has exactly one preimage in each ball. At each such point the local degree is $+1$ or $-1$ as chosen: an orientation-preserving chart restricts the local generator unchanged, and reflection reverses it. F1 gives degree $k\mathrm{sgn}(d)=d$. This works for $k=1$ and for one-dimensional balls as well.