The number of points where the function
f(x) = { |2x² - 3x - 7|, if, x ≤ - 1; [4x² - 1], if, - 1 < x < 1; |x + 1| + |x - 2|, if, x ≥ 1
[t] denotes the greatest integer ≤ t, is discontinuous is _____________.

Solution

Maths sir · 4 steps

The box function breaks only where 4x²-1 crosses an integer. Find those points, then check both joins.

1. Outer pieces. For x ≤ -1 and x ≥ 1, the pieces are moduli of polynomials, so they are continuous inside their own intervals. Trouble can only come from the middle piece or from the joins at x = -1 and x = 1.
2. Middle piece. On -1 < x < 1, 4x² - 1 takes values in [-1, 3). Its least value is -1, at x = 0. The greatest integer function jumps wherever 4x² - 1 crosses the integers 0, 1 or 2.

   4x² - 1 ∈ {0, 1, 2} ⇒ x = ±1/2, ±1/√2, ±√3/2

3. Six inner points. That gives 6 points of discontinuity inside (-1, 1). At x = 0 the value -1 is only touched, not crossed: on both sides 4x² - 1 lies just above -1, so [4x² - 1] = -1 there and nothing jumps.

   Point	Left limit	Value f	Right limit	Verdict
   x = -1	|2 + 3 - 7| = 2	2	4x² - 1 → 3⁻, so 2	continuous
   x = 1	4x² - 1 → 3⁻, so 2	|2| + |-1| = 3	3	discontinuous

4. So. The function is discontinuous at the 6 inner points and at x = 1. That makes 6 + 1 = 7 points.

The common slip: 8. Assuming both joins break, without checking the limits at x = -1, gives 8. Both one-sided limits there equal 2, so x = -1 is a point of continuity.

The idea to keep. For [g(x)], mark where g(x) crosses an integer. At a minimum or maximum, the integer is only touched, so the function does not jump there. Always compute both one-sided limits at every join.

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