Exercise 3, § 2

a) Show that, if $f$ is a continuous mapping of $S_1$ into $\mathbf{R}^n$, $f$ can be extended to a continuous mapping of $B_2$ into $\mathbf{R}^n$ [map the point $t x \in B_2$ ($t \geq 0, x \in S_1$) to $t f(x)$].

b) Deduce that, if $f$ is a continuous mapping of $S_1$ into $S_n$ such that $f(S_1) \neq S_n$, then $f$ can be extended to a continuous mapping of $B_2$ into $S_n$ [use a stereographic projection whose vertex does not lie in $f(S_1)$]. (*)