A monotonically irreducible map of a reducible continuum onto

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.