Continuity

To determine whether the function is continuous at , we need to evaluate the left-hand limit, right-hand limit, and the function value at this point. The greatest integer function , also known as the floor function, returns the largest integer less than or equal to the given number.

1. Calculate the left-hand limit as :
   For slightly less than , is slightly more than 0 (but less than 1), so . Meanwhile, is 0 when is slightly less than , so:
   
2. Calculate the right-hand limit as :
   For slightly greater than , is slightly less than 0, so . Meanwhile, is still 0 when is slightly greater than , so:
   
3. Evaluate at :
   At , and , hence:
   f(\frac{1}{2}) = 0 + 0 = 0
4. Check for continuity at :
   For to be continuous at , the left-hand limit, right-hand limit, and the function value at must all be equal. Here, the left-hand limit is 0 and the right-hand limit is -1, showing they are not equal. Thus, is discontinuous at .

Therefore, the correct answer is is discontinuous at .