In this paper, we first introduce a family S = {$S_n$ : C ${ ightarrow}$ C} of non-Lipschitzian mappings, called total asymptotically nonexpansive (briefly, TAN) on a nonempty closed convex subset C of a real Banach space X, and next give necessary and sufficient conditions for strong convergence of the sequence {$x_n$} defined recursively by the algorithm $x_{n+1}$ = $S_nx_n$, $n{geq}1$, starting from an initial guess $x_1{in}C$, to a common fixed point for such a continuous TAN family S in Banach spaces. Finally, some applications to a finite family of TAN self mappings are also added.