How the relationship between input and output is derived in a comparator?

In the slide 6 of the page about propagation delay of a comparator, the time relationship between input and output is as below. However, I am not sure how the function between input and output voltage is derived. Thank you.

• Perhaps is it measured? – Marko Buršič May 18 '16 at 9:06

The frequency domain transfer function is just a fairly simplified model. It makes a critical assumption that there is a single dominant pole (any other poles are much higher in frequency) and no zeroes, which makes this a first-order model—it can usefully predict real-life behaviour as long as that assumption is true, which it might be for certain circuits. It also specifies $A_v(0)$ is the gain of the comparator at DC, which is another key parameter of the comparator.
The time domain equation corresponds to a unit step function $V_{in} u(t)$ (Laplace: $\frac{V_{in}}s$) applied differentially to the inputs of the comparator. You can obtain the time domain equation by one of: a) multiplying the transfer function by the Laplace transform of the unit step, then taking the inverse Laplace transform; b) taking the inverse Laplace transform of the transfer function (to get the impulse response), then convolving that with the time-domain unit step input. (a is easier.)