Stability of the Cauchy equation on an interval (Q1381549)

The authors study the stability in the sense of Hyers-Ulam for the Cauchy equation on an interval. One of the main results is the following. Theorem. Let \(I\) be either an unbounded or a bounded interval such that \(0\in clI\) and let \(G\) be a subgroup of \(\mathbb{R}\). Then for each \(\delta\geq 0\) and each function \(f:I\cap G\to \mathbb{R}\) satisfying the inequality \[ \bigl | f(x+y) -f(x)-f(y) \bigr| \leq\delta \quad \text{for} \quad x,y, x+y\in I\cap G \tag{*} \] there exists an additive function \(A:G\to \mathbb{R}\) such that \[ \bigl| f(x)-A(x)\bigr| \leq\delta \quad \text{for} \quad x\in I\cap G. \] Thus, under the previous hypotheses, the Cauchy equation is stable in the following sense: there exists a constant \(K\) such that for each \(\delta>0\) and each function \(f:I\cap G\to\mathbb{R}\) satisfying the inequality (*) there exists an additive function \(A:G\to \mathbb{R}\) such that \(| f(x)- A(x)|\leq K\delta\) for \(x\in I\cap G\). The last section of the paper is devoted to estimations from above and from below of the constant \(K\).