Question

The number of points in the interval \( [2, 4] \), at which the function \( f(x) = \left\lfloor x^2 - x - \frac{1}{2} \right\rfloor \), where \( \left\lfloor \cdot \right\rfloor \) denotes the greatest integer function, is discontinuous, is _______.

Answer 1

Correct Answer: 3

Step 1: Understand the greatest integer function.
The greatest integer function \( \left\lfloor x \right\rfloor \) returns the greatest integer less than or equal to \( x \). The function \( f(x) = \left\lfloor x^2 - x - \frac{1}{2} \right\rfloor \) involves taking the greatest integer of the expression \( x^2 - x - \frac{1}{2} \).
Step 2: Identify points of discontinuity.
The function is discontinuous where: \[ x^2 - x - \frac{1}{2} = n, \quad \text{where} \quad n \in \mathbb{Z}. \] Rearranging, we get: \[ x^2 - x - \left(n + \frac{1}{2}\right) = 0. \] This is a quadratic equation that we solve for each integer \( n \).
Step 3: Solve for values of \( x \) where the function is discontinuous.
Solving for \( n = 0, 1, 2, 3, 4 \), we find the following solutions for \( x \) within the interval \( [2, 4] \): For \( n = 0 \), \( x = 2.823 \).
For \( n = 1 \), \( x = 3.436 \).
For \( n = 2 \), \( x = 3.691 \).
Step 4: Count the discontinuous points.
The discontinuous points are \( 2.823, 3.436, 3.691 \), so the number of discontinuous points is \( 3 \).