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Hyers-Ulam stability of mean value points. (English) Zbl 1219.39013
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: $$f'(x)=\frac{f(x)-f(a)}{x-a}\,.$$

39B82Stability, separation, extension, and related topics
26A24Differentiation of functions of one real variable
39B22Functional equations for real functions
Full Text: EMIS EuDML