Let \(S\) be a set, let \(X\), \(Y\) be Banach spaces, and let \(\varepsilon\geq 0\) be arbitrary. We say that a function \(f: S\to X\) is \(\varepsilon\)-onto if \(\forall_{x\in X} \exists_{s\in S}\|x-f(s)\|\leq \varepsilon\). Cleary a given function is \(0\)-onto iff it is a surjection. A Banach space \(X\) is said to have property (D) if every dense subset of \(X\) has the same cardinality as \(X\). A function \(f: X\to Y\) is said to be an \(\varepsilon\)-isometry if \(|\|f(x)- f(y)\|-\|x-y\||\leq \varepsilon\) for \(x,y\in X\). In the present paper the author proves the following results: Theorem 1. Let \(X\) be a Banach space which has property (D) and let \(\varepsilon> 0\). Then for every set \(S\) and every \(\varepsilon\)-onto function \(f: S\to Y\) there exists a surjective function \(F: S\to X\) such that \(\|f(s)- F(s)\|\leq 7\cdot\varepsilon\) for \(s\in S\). Moreover, given a countable subset \(C\subset S\) we can choose \(F\) in such a way that \(F_{|C}= f_{|C}\). We remark that the assumption that \(X\) has the property (D) is essential. Theorem 3. Let \(X\), \(Y\) be Banach spaces, and let \(f: X\to Y\) be an \(\varepsilon\)-isometry which is \(\sigma\)-onto and such that \(f(0)= 0\). Then there exists a unique linear isometry \(U: X\to Y\) such that \(\|f(x)- U(x)\|\leq 2\cdot\varepsilon+ 35\cdot\delta\) for \(x\in X\). If \(\dim X= \dim Y<+\infty\) then this estimation is independent of \(\delta\).