Question

Which of the following options is/are true for the Greatest integer function [.]?

Answer 1

We know that,
The real function $f:R\to R$ defined by $f\left(x\right)=\left[x\right]$ takes value less than or equal to $x$

So,
$\left[x\right]=x$ if $x$ is an integer

We know that, $x-1<\left[x\right]\le x\cdots \left(1\right)$

If we add one in the given inequality $x-1+1<\left[x\right]+1\le x+1$
$\Rightarrow x<\left[x\right]+1\le x+1\cdots \left(2\right)$

From equation $\left(1\right)$ we know that $\left[x\right]\le x$

Now from equation $\left(1\right),\left(2\right)$
$\left[x\right]\le x<\left[x\right]+1$

Now we have, $[x+m]=\left[x\right]+m,\text{if}m\in Z$
We know that the greatest integral value of an integer is equal to the integer itself. And for other numbers except integer, greatest integral value will be $\left[x\right].$ And hence we can write $\left(\right[x+m]$ as sum of greatest integral function and an integer $\left[x\right]+m,\text{where}m\in Z.$