A monotonically irreducible map of a reducible continuum onto [0,1]

We construct a continuum X which is not irreducible, and which admits a monotone map f onto [0,1] such that f is monotonically irreducible in the following sense: For each closed proper subset D of X such that f(D)=[0,1], f restricted to D is not monotone. The example answers a question by L. Mohler and L.E. Ward.