Hyers-Ulam stability of mean value points

We extend and simplify proofs of some recent results on stability in the sense of Hyers and Ulams for mean value points that come from differential expressions, like Rolle theorem and co. Then, classical mean value theorems due to Lagrange and Cauchy are generalized in the setting of divided differences with multiplicities and we prove Hyers-Ulam stability for the corresponding mean value points.