Which of these statements about the floor and ceiling functions are correct?

Statement I: $\lfloor 2x\rfloor=\lfloor x\rfloor+\left\lfloor x+\frac{1}{2}\right\rfloor$ for all real number $x$

Statement II: $\lceil x+y\rceil=\lceil x\rceil+\lceil y\rceil$ for all real numbers $x$ and $y$

Statement I is a standard identity of floor function:

$\lfloor 2x\rfloor=\lfloor x\rfloor+\left\lfloor x+\frac{1}{2}\right\rfloor$

So, Statement I is true.

Statement II is not always true.

Take $x=0.2$ and $y=0.2$.

$\lceil x+y\rceil=\lceil 0.4\rceil=1$

But,

$\lceil x\rceil+\lceil y\rceil=\lceil 0.2\rceil+\lceil 0.2\rceil=1+1=2$

So, Statement II is false.