Question

The number of points, where \( f(x) = \lfloor x \rfloor \), \( 0 < x < 3 \) (\(\lfloor \cdot \rfloor\) denotes the greatest integer function), is not differentiable is:

• A. 1
• B. 2
• C. 3
• D. 4
• E.

Hint:

The greatest integer function \( \lfloor x \rfloor \) is not differentiable at integer points because of the discontinuity in its value at these points.

Answer 1

The Correct Option is B

The function \( f(x) = \lfloor x \rfloor \) (greatest integer function) assigns the greatest integer less than or equal to \( x \) for any real number \( x \). 1. Behavior of \( \lfloor x \rfloor \): - The function \( \lfloor x \rfloor \) is constant in each open interval between consecutive integers, i.e., it takes the same value in \( n \leq x < n+1 \) for any integer \( n \). - However, at integer points \( x = n \), the function has a discontinuity in its derivative because of the jump in its value. 2. Points in the Given Interval \( 0 < x < 3 \): - The interval \( 0 < x < 3 \) contains the integers \( 1 \) and \( 2 \). - At \( x = 1 \) and \( x = 2 \), the function \( f(x) = \lfloor x \rfloor \) is not differentiable due to the discontinuity in its derivative. 3. Number of Non-Differentiable Points: - The function \( f(x) \) is not differentiable at \( x = 1 \) and \( x = 2 \). - Therefore, the total number of non-differentiable points is \( 2 \). Hence, the correct answer is (B) 2.