Identity function and floor function

Question

$f_{3}: \mathbb{Z} \rightarrow \mathbb{Z} \text { with rule } f_{3}(x)=\lfloor x / 2\rfloor+\lceil x / 2\rceil$

Can you please tell me whether this is identity function on its domain ?

I guess yes because $\lfloor x / 2\rfloor$ = $x /2$ $-$ {$x$/2} and same for ceiling function so when we add it gives $x$ and hence the result is true. Can somebody write it clearly for me?

Answer 1

HINT: if $x$ is odd then $\{x/2\}=1/2$ and $\lceil x/2\rceil=\lfloor x/2 \rfloor+1$.

Answer 2

@Masacroso answered your question, But furthermore:

If you think of it on noninteger numbers, it return the Odd number between two even numbers surround $x$. That also can apply to odd integers.