The given function f(x)=[x] is discontinuous at:

The correct answer is Option (1) → every integer

[x] is the greatest integer function: the greatest integer not exceeding x.

At an integer n, the value and the two one-sided limits are:

    \(\text{left-hand limit} = n-1\)
    \(\text{right-hand limit} = n = f(n)\)

The left-hand limit differs from f(n), so f jumps by 1 and is discontinuous at every integer.

At a non-integer c, [x] keeps the same constant value on a small interval around c, so f is continuous there. Hence the points of discontinuity are exactly the integers.

NCERT (2026-27 edition):
Continuity