- az_lenderLv 71 year ago
If f(x) < x for ALL x in [0,1], then the area would be less than 1/2, because the graph of f would lie entirely below the top edge of the triangle described by f(x) = x on [0,1].
If f(x) > x for ALL x in [0,1], then the area would be greater than 1/2, same reasoning.
If f(x) = x for all x in [0,1], there is nothing to prove, we are finished.
The only remaining cases are cases where SOME x's in [0,1] have f(x) less than x, while some other x's in [0,1] have f(x) greater than x. But then the graph of (a continuous) f(x) must cross the line f(x) = x, somewhere.