JEE Main 2026 — Apr 2, Shift 2 — Mathematics solution

JEE Main2026Apr 2, Shift 2Mathematics

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_______.

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Solution

The inner expression $g (x) = x^{2} - x - \frac{1}{2}$ is continuous and strictly increasing on $[2, 4]$ (since $g^{'} (x) = 2 x - 1 > 0$ ).

At the endpoints: $g (2) = 4 - 2 - 0.5 = 1.5$ and $g (4) = 16 - 4 - 0.5 = 11.5$ . So $g$ ranges over $[1.5, 11.5]$ .

The greatest integer function $[g (x)]$ has discontinuities exactly where $g (x)$ takes integer values within this range.

The integers in $[1.5, 11.5]$ are $2, 3, 4, 5, 6, 7, 8, 9, 10, 11$ — a total of $10$ values.

By monotonicity, each integer is attained exactly once, producing $10$ points of discontinuity.

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