Finely continuously differentiable functions (Q5893930)

This paper establishes an explicit characterization of those real-valued functions on a finely open set in Euclidean space which are continuously differentiable with respect to the fine topology of classical potential theory. Differentiability of all orders is also considered, and some consequences of the characterization are deduced. Recall that the fine topology is the coarsest topology on \(\mathbb{R}^n\) which renders all superharmonic functions continuous. A linear mapping \(L:\mathbb{R}^n\to\mathbb{R}\) is called the fine differential of a function \(f\) at a point \(x_0\in U\) if \[ \text{fine lim}_{x\to x_0}\frac{f-f(x_0)-L(x-x_0)}{|x-x_0|}=0. \] Then \(L\) is uniquely determined and will be denoted \(d_{\text{f}}f(x_0)\). A function \(f\) is said to belong to fine-\(C^1(U)\) if \(f\) and \(d_{\text{f}}f\) are both finely continuous. This can then be extended via the standard way to define the class of functions fine-\(C^k(U)\). Finally, for any \(k\in\mathbb{N}\cup\{\infty\}\), the class \(C_{\text{f}-\text{loc}}^k(U)\) is the collection of functions such that, for each \(x\in U\), there is a fine neighborhood \(V\subset U\) of \(x\) and a function \(\overline{f}\in C^k(\mathbb{R}^n)\) such that \(f=\overline{f}\) on \(V\). The main result of the paper is that, for each finely open set \(U\subset\mathbb{R}^n\) and all \(k\in\mathbb{N}\cup\{\infty\}\), fine-\(C^k(U)=C_{\text{f}-\text{loc}}^k(U)\). The author then uses this result to provide several interesting corollaries. Corollary 2 answers a question of \textit{R. Lávička} [Adv. Appl. Clifford Algebr. 17, No. 3, 549--554 (2007; Zbl 1131.31006)], by showing that if \(U\subset\mathbb{R}^n\) is a fine domain and a function \(f:U\to\mathbb{R}\) has zero fine differential at every point of \(U\), then \(f\) is constant.