Continuous differentialble functions

X= C^1 ( (0,1) ) = { f s.t. the continuous derivative f^(1) exists on (0,1) }
Y= C^1 ([0,1]) = { f s.t. the continuous derivative f^(1) exists on [0,1]}
Z= C^0 ([0,1]) = { the set of all continuous functions defined on [0,1] }
Suppose f∈ X∩Z and there exist F∈Z such that F= f^(1) on (0,1) , then
is it necessary f∈Y ???

(I have posted this question before and maybe I was too ambitious. I am going to invite anybody who could do this question to try my perious question )