Summary: Let \(X, Y\) be locally compact Hausdorff spaces, and let \(C_0(X), C_0(Y)\) be the Banach spaces of all real continuous functions on \(X, Y\), respectively, vanishing at infinity endowed with the usual sup-norm. Suppose that \(A, B\) are subspaces of \(C_0(X), C_0(Y)\), strongly separating points of \(X, Y\), respectively; and denote by \(\partial A, \partial B\) the Shilov boundary of \(A, B\), respectively. If \(T\) is a standard \(\varepsilon\)-isometry from \(A\) into \(B\) for some \(\varepsilon > 0\), we prove that there is a non-empty subset \(Y_0\) of \(\partial B\), a surjective continuous map \(\tau : Y_0\rightarrow \partial_0 A\) (where \(\partial_0 A=\partial A \cap \mathrm{supp}(A)\)) and \(h\in C(Y_0)\) with \(|h(y)|=1\) such that \[ |h(y)f(\tau (y))-(Tf)(y)|\leq 3\varepsilon \] for all \(y\in Y_0\) and \(f\in A\). We also give an example to show the constant 3 in the above inequality is optimal.