Question

Let [x] be the greatest integer ≤ x. Then the number of points in the interval (-2,1), where the function f(x)=|[x]|+√x−[x] is discontinuous is _____.

Answer 1

Correct Answer: 2

To find the number of points where the function \( f(x)=|\lfloor x \rfloor|+\sqrt{x}-\lfloor x \rfloor \) is discontinuous in the interval \((-2, 1)\), we analyze each component of the function: the floor function \(\lfloor x \rfloor\), which causes discontinuity at integer points, and \(\sqrt{x}\), which is continuous for \(x \geq 0\).

1. **Interval and Floor Function Analysis:**
(a) The greatest integer function \(\lfloor x \rfloor\) is discontinuous at integer points. Since we consider the greatest integer less than or equal to \(x\), the points of discontinuity for \(\lfloor x \rfloor\) in \((-2, 1)\) are \(x = -2, -1, 0, 1\). However, the interval is \((-2, 1)\), thus we exclude \(x = -2\) and \(x = 1\), focusing only on possible discontinuities at \(x = -1\) and \(x = 0\).

2. **Absolute Value and Square Root Functions:**
(a) The function \(|\lfloor x \rfloor|\) shifts at points where \(\lfloor x \rfloor\) changes value. In this range, \(|\lfloor x \rfloor|\) shifts at \(x = -1, 0\), but this doesn't inherently create new discontinuities beyond those caused by \(\lfloor x \rfloor\).
(b) \(\sqrt{x}\) is continuous for \(x \geq 0\), thus it doesn't contribute additional discontinuities in \((-2, 1)\).

3. **Discontinuity Count:**
(a) Account for discontinuities solely from changes in \(\lfloor x \rfloor\):
  - At \(x = -1\): \(\lfloor x \rfloor\) shifts from \(-1\) to \(0\), causing a discontinuity.
  - At \(x = 0\): \(\lfloor x \rfloor\) shifts from \(0\) to \(1\), causing another discontinuity.

4. **Conclusion:**
The function is discontinuous at \(x = -1\) and \(x = 0\) within the interval \((-2, 1)\). Therefore, there are 2 points of discontinuity.

The computed number of discontinuous points is 2, which fits within the expected range \(2, 2\).