I stumbled upon this theorem after trying to come up with a mapping of color space, RGB to RGB, where each color maps to a new one. (It was for a mouse cursor effect in a painting app.) I couldn't come up with one, and indeed it is impossible!
Are there any constraints on the mapping? If no, then why not map each intensity value to the immediately next value, and then circle it around? For instance, 0 -> 1, 1 -> 2, 2 -> 3, ...., 255 -> 0.
The function you are describing is not continuous, so the theorem does not apply.
But yeah, it has no fixed point.