There two question. The first one is
If $a$ and $b$ are two pure rationalizable strategies, can 0.5$a$+0.5$b$ fail to be a rationalizable strategy?
Yes. Consider the example given
\begin{tabular}{c|c|c|c|} \multicolumn{1}{c}{} & \multicolumn{1}{c}{$b_1$} & \multicolumn{1}{c}{$b_2$} & \multicolumn{1}{c}{$b_3$} \\ \cline{2-4} $a_1$ & 6,9 & 6,3 & 7,2 \\ \cline{2-4} $a_2$ & 5,4 & 5,6 & 5,5 \\ \cline{2-4} $a_3$ & 8,3 & 0,2 & 2,2 \\ \cline{2-4} \end{tabular}
$$ A(h) \cdot a(0) = \left( \begin{array}{ccc} 1 \\ 1 \\ 1+h \end{array} \right), $$