Which of the following functions is periodic?
(a) where denotes the greatest integer less than or equal to the real number
(b) for ,
(c)
(d) None of the above
(a) where denotes the greatest integer less than or equal to the real number
(b) for ,
(c)
(d) None of the above
- A.where denotes the greatest integer less than or equal to the real number
- B.for ,
- C.
- D.None of the above
Show answer and solution
Correct option: A
Solution
f(x) = x - [x] is the fractional part function, which equals x minus its integer part. For any integer n, f(x + n) = (x + n) - [x + n] = (x + n) - ([x] + n) = x - [x] = f(x), so it repeats with period 1. The other functions fail: sin(1/x) is not periodic because its oscillations bunch up near 0 and cannot satisfy f(x+T) = f(x) for all x including x near 0; x·cos x grows in amplitude so no fixed T can work since the amplitude at x+T differs from that at x.
[1983]