  We show that for group actions on locally connected spaces the maximal
equicontinuous factor map is always monotone, that is, the preimages of single
points are connected. As an application, we obtain that if the maximal
continuous factor of a homeomorphism of the two-torus is minimal, then it is
either (i) an irrational translation of the two-torus, (ii) an irrational
rotation on the circle or (iii) the identity on a singleton.
