Approximate continuity points of derivatives of functions of several variables (Q1911354)

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, \textit{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 \(A_{\nabla F}\).