Hyers--Ulam stability of derivations and linear functions

TL;DR

This paper investigates the stability of derivations and linear functions under approximate conditions, showing that functions nearly satisfying derivation rules can be closely represented as sums of derivations and linear functions.

Contribution

It establishes new stability results for derivations along algebraic curves, extending the understanding of Hyers-Ulam stability in this context.

Findings

Approximate derivations are close to exact sums of derivations and linear functions.

Stability theorems are proved for functions with bounded Cauchy differences.

Results apply to functions satisfying derivation rules along algebraic curves.

Abstract

In this paper the following implication is verified for certain basic algebraic curves: if the additive real function $f$ approximately (i.e., with a bounded error) satisfies the derivation rule along the graph of the algebraic curve in consideration, then $f$ can be represented as the sum of a derivation and a linear function. When, instead of the additivity of $f$, it is assumed that, in addition, the Cauchy difference of $f$ is bounded, a stability theorem is obtained for such characterizations of derivations.