Solution
= Solution
\b[No such map exists.] If $H$ fixed the boundary, it would satisfy the hypothesis of part (iii) for every $\varepsilon>0$. Choose $0<\varepsilon<1-|q|$. Then part (iii) would give $q\in H(B)$, contradicting the stipulated omission of $q$. Equivalently, radial projection from the omitted point would turn $H$ into a forbidden <retraction>.