Answer
Rolle's Theorem does not apply (f(x) is not continuous on $[0,1]$), so there are no contradictions.
Work Step by Step
The hypotheses for Rolle's Theorem are
(1) $y=f(x)$ is continuous over the closed interval $[a, b]$ and
(2) $f(x)$ is differentiable at every point of its interior $(a, b)$.
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In this problem,
(1) - the only issue with continuity could arise at $x=1.$
$\displaystyle \lim_{x\rightarrow 1^{-}}f(x)=\lim_{x\rightarrow 1^{-}}x=1$
Since the left limit at $x=1$ does not equal $f(1)=0,$
$f(x)$ is not left-continuous at the right border of $[0,1]$;
that is, f is not continuous on $[0,1]$.
Thus, hypothesis (1) is not satisfied.
