Question

Let \( f(x) = x - \lfloor x \rfloor \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function and \( x \in (-1, 2) \). The number of points at which the function is not continuous is:

• A. 1
• B. 2
• C. 3
• D. 4
• E. 0

Hint:

The greatest integer function is discontinuous at integer points. Count the integer points in the given range for discontinuities.

Answer 1

The Correct Option is B

The greatest integer function \( \lfloor x \rfloor \) is discontinuous at integer values of \( x \), because it "jumps" as \( x \) crosses an integer. 
For \( f(x) = x - \lfloor x \rfloor \), the function is continuous except at integer values of \( x \), because at integer points, the greatest integer function causes a discontinuity. 
Given \( x \in (-1, 2) \), the points where the function is not continuous are at \( x = 0 \) and \( x = 1 \), as these are integer points within the interval. 
Thus, there are 2 points of discontinuity.