If $\left[x\right]$ and $\left[x\right]$ represent integral and fractional part of x then the function defined by $f\left(x\right)=\left[x\right]+$$\sum _{r=1}^{1000} \frac{\{x+r\}}{1000}$ is equal to 

If $x\in \mathbf{I}\text{ then }\left[x\right]=x\text{ and }\{x+r\}=0$ for any $r\in \mathbf{I}\text{. Thus }f\left(x\right)=x\text{. If }x\in \mathbf{R}~\mathbf{I}\text{ then }\left[x\right]=$ integral part $x\text{ and }\{x+r\}=\left\{x\right\}\text{ for any }r=1,2,\cdots 1000\text{. }$Thus $f\left(x\right)=\left[x\right]+\left\{x\right\}=x$