Rolle's Theorem and Piece Wise Functions

Question

The piece wise function f(x)=  0    ,   x=0           is differentiable on (0,1) and satisfies f(0)=f(1), but its derivative is never 

                                           1-x  ,   0<x≤1

 

zero on (0,1). Can Rolle's theorem apply to the function, or is it contradicted?

Answer 1

Rolle's theorem cannot be applied because the function is not continuous at 0.  The limit as x tends to 0 of 1 - x is 1, while f(0) = 0.