Question

Prove that the greatest integer function defined by $f\left(x\right)=\left[x\right],0<x<3$ is not differentiable at $x=1$ and $x=2$.

Answer 1

$f\left(x\right)=\left[x\right]$ is discontinuous at all integral points.

As $x\in (0,3)$, the integers in this interval are $x=1$ and $x=2$.

Hence, at these points the function is discontinuous.

As, the function is discontinuous, it is non differentiable.