Greatest Integer Function

The greatest integer function [x] returns the largest integer less than or equal to x, behaving differently for negative non-integers, a subtlety JEE Advanced frequently probes. Evaluate each term: [2.7] = 2 since 2 ≤ 2.7 < 3. For [-2.7], the greatest integer not exceeding -2.7 is -3, not -2, because -3 ≤ -2.7 < -2. Next [0.4] = 0 since 0 ≤ 0.4 < 1, and [-0.4] = -1 because -1 ≤ -0.4 < 0. Summing gives 2 + (-3) + 0 + (-1) = -2. Option 0 wrongly assumes [-x] = -[x]. Option -1 mishandles the [-0.4] term. Option 1 treats negative arguments like positive ones. Hence the sum equals -2. Plausibility check: for any non-integer x the identity [x] + [-x] = -1 holds, so [2.7]+[-2.7] = -1 and [0.4]+[-0.4] = -1, summing to -2, exactly matching the direct computation and confirming the step-function behavior. The greatest integer function carries a jump discontinuity at every integer, which is why it appears so often in limit, continuity and definite-integral problems on the JEE Advanced paper. Internalizing the identity that [x]+[-x] equals minus one for every non-integer x converts otherwise tedious case evaluations into rapid sign bookkeeping that can be completed almost by inspection.