The object is to determine what
theorems for single-valued functions can be extended to which class of multi-valued
functions. It is shown that an arc cannot be mapped onto a circle by a continuous,
monotone multi-valued function when the image of each point is an arc. On the other
hand, the arc can be mapped onto a nonlocally connected space by a monotone,
continuous function such that the image of each point is an arc. Characterizations
of nonalternating functions analogous to the results in the single-valued
theory are obtained, and it is shown that an nonalternating semi-single-valued
continuous function on a dendrite is monotone. An analog of the monotone
light factorization theorem is obtained for semi-single-valued continuous
functions.
Some other results are: an open continuous function with finite images maps a
regular curve onto a regular curve, and a continuous function with finite images
maps a locally connected, compact space onto a locally connected compact
space.
