  We study unique continuation properties of solutions to the b-family of
equations. This includes the Camassa-Holm and the Degasperi-Procesi models. We
prove that for both, the initial value problem and the periodic boundary value
problem, the unique continuation results found in \cite{LiPo} are optimal. More
precisely, the result established there for the constant $c_0=0$ fails for any
constant $c_0\neq 0$.
