On the mathematics of assertions

Assertions are boolean functions and are sampled at regular time intervals. This currently is the known definition. Boolean algebra doesn't have any notion of continuity. There are only two logical states, which are true and false. Time is not a discrete quantity. It's a continuously increasing quantity. Digital circuits can be modeled as boolean functions. But the state of these functions cannot be determined with reference to a continuously varying quantity. Because in that case there is a violation of the algebraic properties of boolean functions. I don't have sufficient mathematics in place to prove this. The only way that seems reasonable is either to model time in terms of a boolean quantity or convert the resulting boolean function into a decimal function. The domain of comparison must be same. Its the same way that we cannot compare an integer with a complex number. The imaginary part adds additional attribute to it. For the time being there is a tendency to rely heavily on the accuracy of assertions for the functional correctness of a digital circuit. But aren't we missing something?