Question

The period of $f(x) = x - \lbrack x\rbrack,$ if it is periodic, is

• A. $f(x)$ is not periodic
• B. $\frac{1}{2}$
• C. 1
• D. 2

Answer 1

Answer: (A)

$f(x)$ is not periodic

Answer 2

Let $f(x)$ be periodic with period T. Then,

$f(x + T) = f(x)$ for all $x \in R$ ⇒ $x + T - \lbrack x + T\rbrack = x - \lbrack x\rbrack$ for all $x \in R$ ⇒ $x + T - x = \lbrack x + T\rbrack - \lbrack x\rbrack$

⇒ $\lbrack x + T\rbrack - \lbrack x\rbrack = T$ for all $x \in R$ ⇒ $T = 1,2,3,4,........$

The smallest value of T satisfying,

$f(x + T) = f(x)$ for all $x \in R$ is 1.

Hence $f(x) = x - \lbrack x\rbrack$ has period 1.