  In this paper, we prove that the existence of an $\varepsilon$-isometry from
a separable Banach space $X$ into $Y$ (the James space or a reflexive space)
implies the existence of a linear isometry from $X$ into $Y$. Then we present a
set valued mapping version lemma on non-surjective $\varepsilon$-isometries of
Banach spaces. Using the above results, we also discuss the rotundity and
smoothness of Banach spaces under the perturbation by $\varepsilon$-isometries.
