At \( x=2 \), \( f(x)=[x] \) is
[where [.] is greatest integer function]
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Neither continuous nor differentiable
The one-sided limits of \( [x] \) at x = 2 are
\( \displaystyle \lim_{x\to 2^-}[x]=1 \)
\( \displaystyle \lim_{x\to 2^+}[x]=2 \)
\( \displaystyle f(2)=[2]=2 \)
The left-hand limit is not equal to the right-hand limit, so f is not continuous at x = 2.
A function that is differentiable at a point must be continuous there. Since f is not continuous at x = 2, it cannot be differentiable there either.
Why the other options are incorrect
Option (1) Continuous but not differentiable — Incorrect. \( [x] \) jumps from 1 to 2 at x = 2, so it is not continuous there.
Option (2) Differentiable but not continuous — Incorrect. This can never happen: differentiability at a point always implies continuity at that point.
Option (3) Continuous as well as differentiable — Incorrect. \( [x] \) is neither continuous nor differentiable at any integer.