The function \(f(x) = [x]\), where [x] denotes the greatest integer function, is continuous at \(x =\)
(A) 2.9
(B) 5
(C) −3
(D) 6.5
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (D) only
The greatest integer function jumps by 1 at every integer and is constant between two integers. So it is discontinuous at every integer and continuous at every other point.
Checking the statements
(A) 2.9 — Correct. Near 2.9, \([x] = 2\), a constant.
(B) 5 — Incorrect. The left-hand limit is 4, but \(f(5) = 5\).
(C) −3 — Incorrect. The left-hand limit is −4, but \(f(-3) = -3\).
(D) 6.5 — Correct. Near 6.5, \([x] = 6\), a constant.
NCERT (2026-27 edition):
Continuity
