Let X and Y be real Banach spaces and ${varepsilon};>;0$, p > 1. Let f : $X;{ o};Y$ be a bijective mapping with f(0) = 0 satisfying $$|;{parallel}f(x)-f(y){parallel}-{parallel}{x}-y{parallel};|;{leq}{varepsilon}{parallel}{x}-y{parallel}^p$$ for all $x;{in};X$ and, let $f^{-1};:;Y;{ o};X$ be uniformly continuous. Then there exist a constant ${delta};>;0$ and N(${varepsilon},p$) such that lim N(${varepsilon},p$)=0 and a unique surjective isometry I : X ${ o}$ Y satisfying ${parallel}f(x)-I(x){parallel}{leq}N({varepsilon,p}){parallel}x{parallel}^p$ for all $x;{in};X;with;{parallel}x{parallel}{leq}{delta}$.