= Solution
The <Brouwer fixed-point theorem> says that every continuous self-map of a closed ball in $\mathbb R^n$ has a <fixed point>.
Suppose $g$ has no zero in $\overline B_R$. Then
$$
T(x)=-R\frac{g(x)}{|g(x)|}
$$
is a continuous map from $\overline B_R$ into its boundary, hence into $\overline B_R$. The <Brouwer fixed-point theorem> gives $T(x_0)=x_0$. This forces $|x_0|=R$ and
$$
g(x_0)\cdot x_0=-R|g(x_0)|<0,
$$
contradicting the assumed boundary inequality. Therefore $\boxed{g(x_0)=0\text{ for some }|x_0|\leq R}$. This is the <outward vector field zero theorem>.
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