Continuous functions that are locally constant on dense sets (Q1908098)

For a space \(X\), let \(E_0(X)\) be the family of all continuous real-valued functions on \(X\) that are locally constant on a dense (open) subset of \(X\). It was a question of S. Sidney whether \(E_0(X)\) always separates points of \(X\) for a compact Hausdorff space \(X\). The authors present an example of a path-connected compact Hausdorff space \(X\) which answers Sidney's question in the negative.