<s>[INST] Let \(X\) be a topological space such that for every \(x \in X\) there is an open set \(U\) containing \(x\) such that \(U\) is homeomorphic to \(\mathbb{R}^n\) (the Euclidean space with the standard topology) for some \(n\). Such a space is called locally Euclidean. Define a map \(f: X \to \mathbb{R}\) by \(f(x) = n\) if there is an open set \(U\) containing \(x\) such that \(U\) is homeomorphic to \(\mathbb{R}^n\). Show that \(f\) is locally constant, i.e., for any \(x \in X\), there is a neighborhood \(V \subset X\) of \(x\) such that \(f|_V\) is constant.[/INST]