Let A := (0,1) and let k : A → R be defined as follows. For x ∈ A, x irrational, we define k(x) = 0; for x ∈ A rational and of the form x = m/n with natural numbers m, n having no common factors except 1, we define k(x) := n. Prove that k is unbounded on every open interval in A. Conclude that k is not continuous at any point of A.