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


Given: f(x) =[x], 0 < x <3

because a function f is differentiable at a point x=c in its domain if both its limits as:


are finite and equal.


Now, to check the differentiability of the given function at x=1,


Let we consider the left-hand limit of function f at x=1





because, {h<0=> |h|= -h}



Let we consider the right hand limit of function f at x=1






= 0


Because, left hand limit is not equal to right hand limit of function f at x=1, so f is not differentiable at x=1.


Let we consider the left hand limit of function f at x=2






= =


Now, let we consider the right hand limit of function f at x=2






= 0


Because, left hand limit is not equal to right hand limit of function f at x=2, so f is not differentiable at x=2.


10
1