# 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.

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