Function f(x)=[x] , greatest integer function is(A) Continuous at x=2…

The correct answer is Option (2) → (A) and (D) only

[x] is constant between consecutive integers and jumps at every integer. So it is continuous and differentiable at every non-integer point, and neither at an integer.

So [x] is continuous and differentiable at the non-integers 2.5 and −1.2, and neither continuous nor differentiable at the integers 1 and 2.

Checking the statements

(A) Continuous at \( x=2.5 \) — Correct. 2.5 is not an integer; [x] = 2 on the whole interval [2, 3).

(B) Continuous at \( x=1 \) — Incorrect. Just before 1, [x] = 0, but at x = 1, [x] = 1; the graph jumps.

(C) Differentiable at \( x=2 \) — Incorrect. [x] jumps from 1 to 2 at x = 2, so it is not even continuous there.

(D) Differentiable at \( x=-1.2 \) — Correct. −1.2 is not an integer; [x] = −2 on [−2, −1), so its derivative there is 0.