Ordinary Differential Equations

An absolutely continuous function x(t) on ℝ is a solution, in the sense of Carathéodory, of the problem x′ = f(x), x(0) = x₀ , if for almost every t ∈ ℝ, x′(t) = f(x(t)) and x(0) = x ₀. Show that the problem x′ = ● _Q(x), x(0) = a, where 1 _Q is the characteristic function of the set of rational numbers, has a unique solution if a is irrational, and has no solutions if a is rational.