f(x)= x, 0<and equal to x<1

Question

= 0, x=1

is zero at x=0 and at x=1. it derivative is equal to 1 at every point between 0 and 1, so f' is never zero between 0 and 1, and the graph of f has no tangent parallel to the chord from (0,0) to (1,0).

why this does not contradict the Mean Value Theorem?