Concerning the possible values of upper and lower derivatives (Q1282513)

A mapping \(B\) defined on \(\mathbb{R}^n\) is said to be a differential basis in \(\mathbb{R}^n\) if for each \(x\in \mathbb{R}^n\), \(B(x)\) is a family of open bounded sets containing the point \(x\) such that there exists a sequence \(\{R_k\}\subset B(x)\), \(\text{diam }R_k\to 0\), \(k\to\infty\). For \(f\in L_{\text{loc}}(\mathbb{R}^n)\), the numbers \[ \overline D_B(\textstyle{\int} f,x)= \varlimsup(R\in B(x), \text{diam }R\to 0){1\over | R|}\textstyle{\int}_R f \] and \(\underline D_B(\int f,x)\), obtained from \(\overline D_B(\int f,x)\) by replacing \(\varlimsup\) with \(\varliminf\), are defined. A basis \(B\) is called a Busemann-Feller basis (BF-basis) if for each \(R\in\overline B= \bigcup_{x\in\mathbb{R}^n} B(x)\) we have \(R\in B(y)\) for any \(y\in R\). A basis \(B\) is said to be translation-invariant if \(B(x)= \{x+ R: R\in B(0)\}\), \(x\in\mathbb{R}^n\), where \(0\) is the origin in \(\mathbb{R}^n\). By \(B_2\) we denote a basis in \(\mathbb{R}^2\) for which \(B(x)\) consists of all two-dimensional intervals containing the point \(x\), \(x\in\mathbb{R}^2\). We write \(I^n= (0,1)^n\) and \(f\in L(I^n)\) if \(f\in L(\mathbb{R}^n)\) and \(\text{supp }f\subset I^n\). Theorem 1. There exists a translation-invariant BF-basis \(B\subset B_2\) for which there exists a function \(f\in L(I^2)\), \(f\geq 0\), such that \(f(x)< \overline D_B(\int f,x)<\infty\) almost everywhere on \(I^2\). Theorem 2. There exists a BF-basis \(B\subset B_2\) for which there exists a function \(f\in L(I^2)\), \(f\geq 0\), such that \(f(x)<\underline D_B(\int f,x)= \infty\) almost everywhere on \(I^2\).