If $f(x)= \begin{cases}x[x]& , ~ 0 \leq x<2 \\ (x-1)[x] & ,~ 2 \leq x<3\end{cases}$ where [.] denotes the greatest integer function, then

both f'(1) and f'(2) do not exist

We have,

$f(x)= \begin{cases}x \times 0=0 & ,~~ 0 \leq x<1 \\ x \times 1=x & ,~~ 1 \leq x<2 \\ (x-1) \times 2=2(x-1)& ,~~ 2 \leq x<3\end{cases}$

Clearly, f(x) is not continuous at x = 1. So, it can not be differentiable at x = 1.

We observe that f(x) is continuous at x = 2 but it is not differentiable at x = 2.

Hence, both f'(1) and f'(2) do not exist.