Prove that the greatest integer function defined by f (x) = [x], 0

Question

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

Philoid · Accepted Answer

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.