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Greatest Integer Function Discontinuities on [2,4]: 10 Points | JEE 2026

JEE Maths question with a full step-by-step solution.

Question
The number of points in the interval [2,4][2,4], at which the function f(x)=[x2−x−12]f(x)=\left[x^2-x-\dfrac12\right], where [⋅][\cdot] denotes the greatest integer function, is discontinuous, is
Solution
Answer: 10 (± 0.01)
Step 1: On x∈[2,4]x\in[2,4], let g(x)=x2−x−12g(x)=x^2-x-\dfrac12. It is increasing there, with
g(2)=4−2−12=1.5,g(4)=16−4−12=11.5,g(2)=4-2-\tfrac12=1.5,\qquad g(4)=16-4-\tfrac12=11.5,
 so the range of gg is [1.5, 11.5][1.5,\ 11.5]. Step 2: The greatest integer function [g(x)][g(x)] is discontinuous exactly where g(x)g(x) takes an integer value. Step 3: The integers in the interval [1.5,11.5][1.5,11.5] are
2,3,4,5,6,7,8,9,10,11,2,3,4,5,6,7,8,9,10,11,
 which is 1010 values. Since gg is strictly increasing, each is attained exactly once. Step 4: Therefore the number of points of discontinuity is 1010. Correct answer: 10
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