For the greatest integer function \( f(x)=[x] \), which of the following statements are correct?
A. It is continuous at all real numbers.
B. It is continuous at \( x=2 \) and 3.
C. It is continuous and differentiable for \( x\in(2,3) \)
D. It is continuous and differentiable at \( x=4.5 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → C and D only
The greatest integer function is constant between consecutive integers and jumps at every integer.
So \( [x] \) is discontinuous at every integer, and continuous and differentiable at every non-integer.
Checking the statements
(A) It is continuous at all real numbers — Incorrect. It is discontinuous at every integer.
(B) It is continuous at \( x=2 \) and 3 — Incorrect. At x = 2, \( \lim_{x\to2^{-}}[x]=1 \) but \( f(2)=2 \); the same happens at 3.
(C) It is continuous and differentiable for \( x\in(2,3) \) — Correct. There \( [x]=2 \), a constant.
(D) It is continuous and differentiable at \( x=4.5 \) — Correct. 4.5 lies in (4, 5), where \( [x]=4 \) is constant.