The number of points of discontinuity of the function f(x)=[x²/2]-[√x], x ∈[0,4], where [·] denotes the greatest integer function, is ________.

Solution

Maths sir · 5 steps

List every point where either bracket jumps, then check whether the two jumps cancel anywhere.

1. Where brackets jump. [g(x)] can only break where g(x) crosses an integer. So find where x²/2 is an integer and where √x is an integer, for x ∈ [0,4].
2. First bracket. x²/2 runs from 0 to 8. It is an integer when x² = 0, 2, 4, …, 16, that is at x = 0, √2, 2, √6, √8, √10, √12, √14, 4. Inside (0,4) that gives 7 points.
3. Second bracket. √x runs from 0 to 2. It is an integer at x = 0, 1, 4. Inside (0,4) that gives only x = 1, which is not in the first list, so nothing cancels there. All 8 interior points are discontinuities.
4. Check the endpoints. At x = 0: f(0) = 0 and just to the right both brackets are still 0, so f is continuous. At x = 4: f(4) = 8 - 2 = 6, and just to the left f = 7 - 1 = 6. Both brackets drop by 1 together and the jumps cancel, so f is continuous at 4.
5. So. f is discontinuous at √2, 2, √6, √8, √10, √12, √14 and 1, which is 8 points.

The common slip: 9. Counting x = 4 as a discontinuity gives 9. There both brackets fall by 1 together, so the left limit and f(4) are both 6.

The idea to keep. For a sum or difference of greatest integer functions, list all the jump points first. Then test any shared points and the endpoints one by one, because two jumps can cancel.

Answer from the NTA final answer key. Solution by PrepWiser.