Question

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

Answer 1

Correct Answer: 2

To determine where the function \( f(x) = |[x]| + \sqrt{x-[x]} \) is discontinuous on the interval \((-2,1)\), we need to analyze its components: \( [x] \) (the greatest integer function) and \( \sqrt{x-[x]} \). The greatest integer function, \([x]\), changes at integer values. Specifically, within any interval \((n, n+1)\) where \(n\) is an integer, \([x] = n\).

The discontinuity points can occur where \([x]\) changes value or where \(\sqrt{x-[x]}\) becomes undefined. Since \(\sqrt{x-[x]}\) is defined for all \(x\) (as \(0 \leq x-[x] < 1\)), we only need to check where \([x]\) changes. This happens at integer values. In the interval \((-2,1)\), these points are at \(x = -1, 0\). 

Now, let's analyze the function's behavior at these points:

• At \(x = -1\), \([x]\) changes from \(-2\) to \(-1\). Thus, \( f(x) \) is discontinuous.
• At \(x = 0\), \([x]\) changes from \(-1\) to \(0\). Thus, \( f(x) \) is also discontinuous.

There are 2 points of discontinuity in the interval \((-2, 1)\), verifying that the count falls within the expected range of 2.

Therefore, the number of points of discontinuity is 2.