Floor and Ceiling Functions

For non-integer $x$ , $\lceil x \rceil = \lfloor x \rfloor + 1$ , therefore

$\begin{aligned} \lceil x \rceil + \lfloor x \rfloor & = 5 \\ \lfloor x \rfloor + 1 + \lfloor x \rfloor & = 5 \\ 2 \lfloor x \rfloor + 1 & = 5 \\ \implies \lfloor x \rfloor & = 2 \end{aligned}$

$\implies 2 < x < 3$ and $AB = 2 \times 3 = \boxed 6$ .

We can use an example to conjure a conclusion: $\lceil 2.5 \rceil + \lfloor 2.5 \rfloor = 3+2= 5$

Given that $2 < 2.5 < 3$ , we can see that any number between these bounds will satisfy the equation above, therefore $A = \boxed{2}$ and $B=\boxed{3}$ and the answer is $2\times 3 = \boxed{6}$