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Approximate symmetric derivatives are uniformly closed. (English) Zbl 0759.26007

Let \(I\) be an open interval and \(f: I\to R\). Let \[ \underline f^ s(a)=\liminf_{h\to 0+} {f(a+h)-f(a-h)\over 2h} \] be the lower symmetric derivative of \(f\) at \(a\). Let \(\alpha\) be a real number, \[ A(\alpha)=\left\{t\in R-\{0\}: {f(a+t)-f(a-t)\over 2t}<\alpha\right\} \] and the lower approximate symmetric derivative \(\underline f^ s_{ap}(a)\) of \(f\) at \(a\) let be the least upper bound of the set \(\{\alpha\in R\): the density of \(A(\alpha)\) is zero at 0}.
Lemma. Let \(f: I\to R\) be a nondecreasing function. Then \(\underline f^ s(a)=\underline f^ s_{ap}(a)\) and \(\overline f^ s(a)=\overline f^ s_{ap}(a)\) for each \(a\in I\).
Theorem. Let \(f: I\to R\) be a nondecreasing function. If \(f\) has the approximate symmetric derivative \(f^ s_{ap}(a)\) at \(a\), then \(f\) has also the symmetric derivative \(f^ s(a)\) at \(a\) and it holds \(f^ s(a)=f^ s_{ap}(a)\).
Furthermore, the author proved that the class of all approximate symmetric derivatives on \(I\) of measurable functions is closed with respect to the uniform convergence.

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26B05 Continuity and differentiation questions
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