Continuity
For the greatest integer function, at how many points in the open interval (0,3) is g(x) = [x] + [-x] discontinuous?
Select the correct option:
Solution
2
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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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- continuity
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
2
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.
This medium difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of continuity. It appeared in the 2025 exam.
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