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, denoted as \( f(x) = \lfloor x \rfloor \), exhibits discontinuities at integer points due to abrupt changes in its value. For instance, at \( x = 2 \), the function yields \( f(x) = 2 \) for \( x \) in the interval \( [2, 3) \). It then abruptly increases to \( 3 \) at \( x = 3 \). Consequently, \( f(x) \) is not continuous at \( x = 2 \). As a result of this discontinuity, the function is also not differentiable at \( x = 2 \).