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

JEE Main · 2026 · 02 Apr 2026 · Shift 2 · Continuity and Differentiability

The number of points in the interval [2,4], at which the function

f (x) = [x^{2} - x - \frac{1}{2}]

, where

[\cdot]

denotes the greatest integer function, is discontinuous, is ____ .

Answer: 10

[2, 4]

g (x) = x^{2} - x - \frac{1}{2}

has

g^{'} (x) = 2 x - 1 > 0

, so

g

increases from

g (2) = 1.5

g (4) = 11.5

. The greatest-integer function

[g (x)]

jumps precisely where

g (x)

takes an integer value; the integers in

(1.5, 11.5]

are

2, 3, 4, 5, 6, 7, 8, 9, 10, 11

, each attained once because

g

is strictly increasing. Thus

f

is discontinuous at

10

points.

JEE Main · 2026 · 02 Apr 2026 · Shift 2 · Integer answer