The greatest integer function, \( f(x) = \lfloor x \rfloor \), returns the largest integer less than or equal to any real number \( x \). 1. Behavior of \( \lfloor x \rfloor \): - \( \lfloor x \rfloor \) remains constant on open intervals between consecutive integers; it has the same value for \( n \leq x<n+1 \) where \( n \) is any integer. - At integer points (\( x = n \)), the function exhibits a discontinuity in its derivative due to a jump in its value. 2. Points within \( 0<x<3 \): - The integers \( 1 \) and \( 2 \) are within the interval \( 0<x<3 \). - At \( x = 1 \) and \( x = 2 \), \( f(x) = \lfloor x \rfloor \) is not differentiable because its derivative is discontinuous. 3. Count of Non-Differentiable Points: - \( f(x) \) is not differentiable at \( x = 1 \) and \( x = 2 \). - Consequently, there are \( 2 \) non-differentiable points. Thus, the answer is (B) 2.