Concept:Greatest Integer Function: (Floor function)The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ? x.The domain of [x] is R and the range is I.Note:Any function is differentiable only if it is continuous.The floor function f(x) = ?x? is differentiable in every open interval between integers, (n, n + 1) for any integer n.Calculation:Given that,y = [x]Statement:1 Its derivative is 0 at x = 0.5We know that the floor function is differentiable at all points except integer points.Hence, y = [x] is differentiable at x = 0.5? y = [0.5] = 0? dy/dx = 0Statement:2 It is continuous at x = 0We know that y = [x] is only continuous in the open interval between integers and discontinuous at all integer values.? Only statement 1 is correct.