[vjqjcfuh] Coloring a sphere

Can the surface of a sphere be colored such that every point is a different color and there are no discontinuities?

Easy: Fully immerse the sphere within an RGB color cube and use the colors intersecting with the sphere surface.  Or, continuously deform the surface of the RGB color cube into a sphere.  Or any portion of the cube of genus 0.

But I suspect it is impossible for coloring only grayscale.

Inspired by needing a surface design for coloring a twisty sphere.