Finitely continuous Hamel functions (Q2501044)

Let \(f:\mathbb{R}^n\to\mathbb{R}^k\) be a function and \(\kappa\leq c\) be a cardinal number. A function \(f\) is a Hamel function if \(f\), considered as a subset of \(\mathbb{R}^{n+k}\), is a Hamel basis for \(\mathbb{R}^{n+k}\) (no distinction is made between a function and its graph). The function \(f\) is called \(\kappa\)-continuous if it can be covered by the union of \(\kappa\) many partial continuous functions from \(\mathbb{R}^n\). The main result is: there exists a Hamel function \(h:\mathbb{R}^n\to\mathbb{R}^k\) which is \((n+2)\)-continuous \((k,n\geq 1)\).