The greatest integer function \(f(x) = [x]\) is differentiable for all values of
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(x \in R-Z\)
At a non-integer x: x lies inside some interval (k, k + 1) with k an integer, and on that interval \([x] = k\) is constant. So f is differentiable there, with \(f'(x) = 0\).
At an integer k: the function jumps:
\(\displaystyle \lim_{x \to k^-}[x] = k - 1,\quad \lim_{x \to k^+}[x] = k\)
So f is not continuous at k, and a function that is not continuous at a point cannot be differentiable there.
Hence [x] is differentiable exactly for \(x \in R - Z\).