We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces. We define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): $K$-monotone dominated and cone-to-cone monotone mappings. First we show some relationships between these classes. Then, we study continuity and differentiability (also in the metric and $w^*$ senses) of mappings in these classes.