Let \(f(x) = [x] + [-x]\), where [t] denotes the greatest integer less than or equal to t. Consider the following statements:
(A) \(f(2) = 0\)
(B) \(f\left(\displaystyle\frac{1}{2}\right) = 0\)
(C) \(\displaystyle\lim_{x \to 2} f(x)\) exists and is equal to −1.
(D) f is continuous at \(x = 2\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
If x is an integer, then [x] = x and [−x] = −x:
\(f(x) = x + (-x) = 0\)
If x is not an integer, say \(n \lt x \lt n + 1\), then \(-n - 1 \lt -x \lt -n\):
\(f(x) = n + (-n - 1) = -1\)
So f = 0 at every integer and f = −1 at every other point; near x = 2 (but not at 2), f is −1.
Checking the statements
(A) \(f(2) = 0\) — Correct. [2] + [−2] = 2 − 2 = 0.
(B) \(f\left(\frac{1}{2}\right) = 0\) — Incorrect. [1/2] + [−1/2] = 0 + (−1) = −1.
(C) \(\lim_{x \to 2} f(x)\) exists and is equal to −1. — Correct. f = −1 on both sides of 2.
(D) f is continuous at \(x = 2\). — Incorrect. The limit −1 is not equal to f(2) = 0.