Question: Let A := (0,1) and let k : A R be defined as follows. For x A, x irrational, we define k(x) =

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.

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