Step-by-step Solution:

To determine where the function \( f(x) = [x]^n \) (where \( [x] \) is the greatest integer less than or equal to \( x \) and \( n \geq 2 \) is an integer) is discontinuous, we analyze the behavior of the greatest integer function. \[1. Greatest Integer Function [x] :\] - The function \( [x] \) is discontinuous at all integer values of \( x \). - For any integer \( k \), as \( x \) approaches \( k \) from the left, \( [x] = k - 1 \), and as \( x \) approaches \( k \) from the right, \( [x] = k \). \[2. Function f(x) = [x]^n :\] - Since \( [x] \) is discontinuous at integers, \( [x]^n \) will also be discontinuous at these points because the discontinuity of \( [x] \) affects \( [x]^n \). - For non-integer values of \( x \), \( [x] \) is continuous, and thus \( [x]^n \) is also continuous. Conclusion: The function \( f(x) = [x]^n \) is discontinuous at all integer values of \( x \). Final Answer: \[ \boxed{D} \]