Given a continuous function from a hollow sphere to itself, which has no fixed-points, construct...?

Question

f(x) is a continuous function from the hollow unit sphere to itself, which has no fixed-points. The function f is ONTO.
Yes, I know, f(x) is a paradoxical object which does not exist. But assume it does. Can you construct from it a continuous function g whose domain and codomain are again the hollow unit sphere, and which also contains no fixed-points, BUT the new function g is not ONTO.

Good luck people :)

Answer 1

I've been trying to think of this for awhile. Obviously, you can't perform a retract onto the unit circle (the map is not defined for the poles). So then I thought of sort of a cross (2 circles) but then the map would not be continuous. Maybe a rotation and then a flip would work, but I am not sure how to write the function. Say you rotate all of the points 90 deg about the z-axis and then flip every thing about the x,y plane. I believe that this would be a continuous map and there shouldn't be any fixed points, but of course this map is onto. Well, I am going to keep thinking about this for awhile.