In which of the following functions is Rolle's theorem applicable ?

• A

  $f(x) = \left\{ \begin{array}{l}
  x,\,\,\,\,\,\,\,0 \le x < 1\\
  0,\,\,\,\,\,\,\,x = 0\,\,\,\,\,\,
  \end{array} \right.on\,\,\left[ {0,1} \right]$

• B

  $f(x) = \left\{ \begin{array}{l}
  \frac{{\sin x}}{x},\,\,\,\,\,\,\, - \pi  \le x < 0\\
  0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0\,\,\,\,\,\,
  \end{array} \right.on\,\,\left[ { - \pi ,0} \right]$

• C

  $f(x) = \frac{{{x^2} - x - 6}}{{x - 1}}\,\,\,\,\,on\left[ { - 2,3} \right]$

• D

  $f(x) = \left\{ \begin{array}{l}
  \frac{{{x^3} - 2{x^2} + 5x + 6}}{{x - 1}},\,\,\,\,if\,\,x \ne 0\,\,\,\,\,\,\\
   - 6,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,x = 1\,\,\,\,\,\,\,\,\,\,\,\,\
  \end{array} \right.on\left[ {-2,3} \right]$