Continuity

Question

For the greatest integer function, at how many points in the open interval (0,3) is g(x) = [x] + [-x] discontinuous?

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The key fact is that for non-integer x the identity [x] + [-x] = -1 holds, while for integer x both floor terms coincide and [x] + [-x] = 0, so the function is a constant -1 punctured by jumps at integers. This piecewise-constant structure means discontinuities can occur only at integer points. Within the open interval (0,3) the integers are 1 and 2 (the endpoints 0 and 3 are excluded), giving exactly two candidate points. At each of x = 1 and x = 2 the value jumps from -1 to 0, so both are genuine discontinuities, totalling 2. Option 0 wrongly assumes the sum is everywhere constant, forgetting the integer exception. Option 1 counts only one integer, perhaps overlooking that the interval contains two. Option 3 mistakenly includes an endpoint or a non-integer. The governing JEE pattern is analysing the floor function's behaviour at integers. Plausibility check: just left of x = 1, g = -1, exactly at x = 1, g = 0, and just right again g = -1, a clear removable-style jump confirming the count of two discontinuities.

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