Question

Which of the following function is periodic:

• A. $f\left(x\right)=x-\left[x\right]$ where $\left[x\right]$ denotes the largest integer less than or equal to the real number $x$
• B. $f\left(x\right)=\sin \frac{1}{x}$ for $x\ne 0,f\left(0\right)=0$
• C. $f\left(x\right)=x\cos x$
• D. none of these

Answer 1

The correct option is A $f\left(x\right)=x-\left[x\right]$ where $\left[x\right]$ denotes the largest integer less than or equal to the real number $x$

Note that $x-\left[x\right]=\left\{x\right\}where\left\{x\right\}$ is a fraction part of x.
Also adding an integer $k$ to $x$ do not change the value of fraction part of x + k. That is for $k=1,$ we have $\{x+1\}=\left\{x\right\}$
Hence, $[f\left(x\right)=f\left(x+1\right)]$
Therefore the function $[f\left(x\right)=x-\left[x\right]]$ is periodic with period $T=1$.