Can the difference of ceiling functions be written as ceiling of the difference or a function of the difference?

Question

I need to figure out if

$\lceil a \rceil - \lceil b \rceil = \lceil a-b \rceil$

This is to see if I can write a certain sum as a convolution for a numerical simulation.

I am also interested in how this changes in the case of different signs.

Even if this is not true, if there a way I can express $\lceil a \rceil - \lceil b \rceil$ purely as a function of the difference $a-b$?

Answer 1

So it is not true that $\lceil a \rceil - \lceil b \rceil = f(a-b)$.

We can prove this by counter example, by choosing two pairs of numbers with the same difference but different values under this 'function'. Take (5.2,3.1) and (5.1,3.0) which both have a difference of 2.1 however,

$\lceil 5.2 \rceil - \lceil 3.1 \rceil = 6-4=2$, $\lceil 5.1 \rceil - \lceil 3.0 \rceil = 6-3=3$.

Thus any such $f$ would be a multi-valued function i.e it is not well-defined.