I see no almost fixed point for the function that is 1 until 0.5 and 0 after.
0.5 is the almost fixed point. Its the point where f(x)−x goes from being positive to negative. If you take a sequence of continuous functions fn(x) that converge pointwise to f(x) then there will exist a sequence yn such that fn(yn)=yn and limn→∞yn=0.5.
That definition makes more sense than the one in the question. :)