Let \(K(a_ n/b_ n)\) and K(ã\({}_ n/\tilde b_ n)\) be two continued fractions, and assume that K(ã\({}_ n/\tilde b_ n)\) converges to a value \~f\(\in {\mathbb{C}}\) (or that the modified approximants \~S\({}_ n(w_ n)\) of K(ã\({}_ n/\tilde b_ n)\) converge to a value \~F\(\in {\mathbb{C}})\). The problem treated is finding sufficient conditions - mainly in terms of \(| a_ n-\tilde a_ n|\) and \(| b_ n-\tilde b_ n|\)- for the convergence of \(K(a_ n/b_ n)\) to a value \(f\in {\mathbb{C}}\) (or for the convergence of its modified approximants \(S_ n(w_ n)\) to a value \(F\in {\mathbb{C}})\). Estimates for \(| f-\tilde f|\) (or \(| F-\tilde F|)\) and the truncation errors are given.