Concerning real functions with values in the Cantor set (Q6599552)

Let \(k\ge 2\) be a positive integer and \(Y\subset\mathbb{R}\). A function \(f\colon \mathbb{R}\to Y\) is called \textit{essentially (\(k\)-)nonconstant} if there exist \(k\) distinct points \(y_1,\ldots,y_k\in Y\) such that \(f^{-1}(y_i)\) has finite positive Lebesgue measure for all \(i\le k\).\N\NLet \(X\) be a topological space and \(Y\) be a locally compact space. It is known that if \(A\) is a dense subspace of \(X\) then every continuous map \(f\colon X\to Y\) is extendable to a function \(\hat{f}\colon X\to Y\) that is continuous at every point \(x\in A\), see Theorem 3.2 in [\textit{C. Costantini} and \textit{A. Marcone}, Topology Appl. 103, No. 2, 131--153 (2000; Zbl 0986.54025)]. In the paper under review the author gives a new proof of this fact under the stronger assumption that \(Y\) is a compact Hausdorff space.\N\NThen he applies this fact in the proof of the following result: For every integer \(k\ge 2\), there exist at least continuum-many essentially \(k\)-nonconstant almost everywhere continuous functions from \(\mathbb{R}\) to the Cantor set \(C\).\N\N\textbf{Reviewer's note. } It seems that for every countable cardinal \(\kappa>1\) one can easily construct a \(2^\mathfrak{c}\) of essentially \(\kappa\)-nonconstant a.e. continuous functions \(f\colon\mathbb{R}\to C\).