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Consider the following statements in respect of the function y = [x], x ∈ (-1, 1) where [.] is the greatest integer function: 1. Its derivative is 0 at x = 0.5 2. It is continuous at x = 0 Which of the above statements is/are correct?

Consider the following statements in respect of the function y = [x], x ∈ (-1, 1) where [.] is the greatest integer function: 1. Its derivative is 0 at x = 0.5 2. It is continuous at x = 0 Which of the above statements is/are correct? Correct Answer 1 only

Concept:

Greatest Integer Function: (Floor function)

The function f (x) = is called the greatest integer function and means greatest integer less than or equal to x i.e ≤ x.

The domain of is R and the range is I.

Statement:1  Its derivative is 0 at x = 0.5

We know that the floor function is differentiable at all points except integer points.

Hence, y = is differentiable at x = 0.5

⇒ y = = 0

⇒ dy/dx = 0

Statement:2  It is continuous at x = 0

We know that y = is only continuous in the open interval between integers and discontinuous at all integer values.

∴ Only statement 1 is correct. 

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