Prove that the greatest integer function defined by`f(x) = [x], 0 < x < 3`is not differentiable at `x = 1 a n d x = 2`.

Explanation

`f(x) = [x]`
`at x = 1`
Lf'(1) `= underset (h to 0) (lim)(f(-h)-f(1))/(-h)`
`= underset (h to 0) (lim)(l[1 - h] -[1])/(-h) = underset(h to 0) (lim)(0-1)/(-h)=infty`
Rf'(1)= `underset (h to 0) (lim)(f(1 +h)-f(1))/(h)`
`= underset (h to 0) (lim)(l[1 + h] -[1])/(+h) = underset(hto 0)(lim)(1-1)/(h)`
` = underset (h to 0) (lim) (0)=0`
`:'` Lf' (1) `ne` Rf'(1)
`:.` F(x) = [x] is not differentiable at x = 1 .
at x = 2
Lf' (2) =`underset (h to 0) (lim)(f(2 -h)-f(2))/(-h)`
`= underset (h to 0) (lim)([2 - h] -[2])/(-h) `
` = underset(hto 0)(lim)(1-2)/(-h)=underset(h to0)(lim)(1)/(h)=infty`
Rf'(2) = `underset (h to 0) (lim)(f(2 + h)-f(2))/(h)`
`= underset (h to 0) (lim)([2 + h] -[2])/(h) `
` = underset(hto 0)(lim)(2-2)/(h)=underset(h to0)(lim)(0)=0`
`:'` Lf' (2) `ne` Rf'(2)
`:.` f(x) is not differentiable at x = 2 . Hence proved .