Finite chainability, locally Lipschitzian and uniformly continuous functions (Q1282209)

Let \(X\) be a metric space and \(Y\subset X\). Authors show that the following conditions are equivalent. (i) \(Y\) is \(X\)-finitely chainable, i.e., for each \(\varepsilon>0\) there are a finite set \(q_1, \ldots, q_k\) and a positive integer \(m_Y\) such that any point of \(Y\) can be bound with some \(q_j\) by an \(\varepsilon\)-chain with the length \(m_Y\). (ii) For any uniformly continuous function \(f\colon X\to \mathbb{R}\), \(f(Y)\) is a bounded subset of \(\mathbb{R}\). (iii) For any uniformly locally Lipschitzian function \(f: X\to \mathbb{R}\), \(f(Y)\) is a bounded subset of \(\mathbb{R}\).