Question

If \( f(x) = \lfloor x \rfloor \) is the greatest integer function, then the correct statement is:

• A. \( f \) is continuous but not differentiable at \( x = 2 \).
• B. \( f \) is neither continuous nor differentiable at \( x = 2 \).
• C. \( f \) is continuous as well as differentiable at \( x = 2 \).
• D. \( f \) is not continuous but differentiable at \( x = 2 \).

Hint:

The greatest integer function \( \lfloor x \rfloor \) has discontinuities at integer values of \( x \), and hence, it is neither continuous nor differentiable at these points.

Answer 1

The Correct Option is B

The greatest integer function \( f(x) = \lfloor x \rfloor \) is not continuous at integer points, because the value of the function jumps at these points. At \( x = 2 \), \( f(x) \) takes the value 2 for \( x \in [2, 3) \) and jumps to 3 at \( x = 3 \). Thus, \( f(x) \) is not continuous at \( x = 2 \), and since the function is not continuous, it is also not differentiable at \( x = 2 \).