Hyers-Ulam stability of mean value points (Q533308)

The authors consider a few problems concerning the stability for Lagrange's and Flett's mean value points. The first result reads as follows. Let \(f:\mathbb{R}\to\mathbb{R}\) be a continuously twice differentiable mapping and let \(\eta\in(a,b)\) be a unique Lagrange's mean value point of \(f\) in \((a,b)\) (i.e., \(f'(\eta)=\frac{f(b)-f(a)}{b-a}\)). It is proved that for each \(\varepsilon>0\) there exists \(\delta>0\) such that for each differentiable function \(g:\mathbb{R}\to\mathbb{R}\) satisfying \(|f(x)-g(x)|\leq\delta\), \(x\in [a,b]\) there exists a Lagrange's mean value point \(\xi\in (a,b)\) of \(g\) such that \(|\xi-\eta|\leq\varepsilon\). Other results are connected with approximate mean value points: \[ \left|f'(\xi)-\frac{f(b)-f(a)}{b-a}\right|\leq\varepsilon \] and with the stability of the equation: \[ f'(x)=\frac{f(x)-f(a)}{x-a}\,. \]