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Approximate continuity points of derivatives of functions of several variables. (English) Zbl 0851.26006

For every function \(f : \mathbb{R}^m \to \mathbb{R}\) denote by \(A_f\) the set of approximate continuity points of \(f\). (If \(m > 1\), then we consider the ordinary differentiation basis in \(\mathbb{R}^m.)\) In 1983, G. Petruska [Acta Math. Hung. 42, 355-360 (1983; Zbl 0542.26005)] verified that if \(F : \mathbb{R} \to \mathbb{R}\) is differentiable and \(f = F'\), then \(f\) takes each of its values on \(A_f\). It is natural to ask if an analogous result holds for functions \(F : \mathbb{R}^m \to \mathbb{R}\). The results: 1. If \(F : \mathbb{R}^m \to \mathbb{R}\) is differentiable, then \(f = \partial F/ \partial x_1\) takes each of its values on \(A_f\). 2. There is a continuous function \(F : \mathbb{R}^2 \to \mathbb{R}\) such that \(f = \partial F/ \partial x_1\) exists everywhere and \(f\) does not take each of its values on \(A_f\). 3. There is a differentiable function \(F : \mathbb{R}^2 \to \mathbb{R}\) whose gradient \(\nabla F\) does not take each of its values on \(A_{\nabla F}\).

MSC:

26B05 Continuity and differentiation questions
26B35 Special properties of functions of several variables, Hölder conditions, etc.

Citations:

Zbl 0542.26005
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References:

[1] G. Petruska, Derivatives take every value on the set of approximate continuity points,Acta Math. Hungar.,42 (1983), 355–360. · Zbl 0542.26005 · doi:10.1007/BF01956783
[2] Queries section,Real Analysis Exchange,16 (1990–91), 373.
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