A set valued intermediate value theorem.

Question

Does anyone know of an intermediate value theorem for set valued maps which are upper semicontinuous? Specifically, I'm looking for a theorem which says that, for example, if we are in $R^3$ and we look at two points and one maps to a set in the first quadrant and the other to a set in the eighth quadrant then on a line between them there either exist a finite amount of points mapping to the quadrants that a path from the first quadrant to the eighth quadrant must go through or there exists a point (on the line) which maps to the origin. Thanks.