Hard to describe! Im thinking something like the cube root function shifted so that its inflection point is at (0.5,0.5).So an appropriate function should be f(x) = cuberoot{x-0.......
When 0 < x < 0.5, we get a value < x but on interval (0, 0.5)
When 0.5 < x < 1, we get value > x but on interval (0.5, 1)
Consider the unit square in the plane. Cut it into four smaller equal sized squares. You want a graph that lies entirely below the bottom diagonal of the bottom left square and above the top diagonal of the upper right square. Hard to describe! I'm thinking something like the cube root function shifted so that its inflection point is at (0.5,0.5).
So an appropriate function should be f(x) = cuberoot{x-0.5} + 0.5
EDIT: We also need to scale the whole thing by a factor of 1/2 so that it stays inside the unit square, so it should be
f(x) = cuberoot{(x-1/2)/2} + 1/4
EDIT 2: Ok that doesn't *quite* do it the right way, but the idea is right, just have to scale it around properly to get it to fit into the unit square. We want a cuberoot shaped function passing through (0,0) and (1,1) with inflection point at (0.5,0.5).
Point is, definitely possible :)
