Digital is a subset of analog.
Digital logic follows three rules:
1. Gates exist. That is, logic elements with well-defined input and output states.
2. The gates operate with a Boolean algebra, utilizing a certain number of signal states.
3. Specified min/max propagation delays. (Propagation is a delay between an input state change and a corresponding output state change.)
Anywhere these characteristics arise in an analog circuit, it can be considered digital as well.
Conversely, it often happens that a circuit, intended to be digital, violates these rules. If the interconnects between gates are not simple propagation delays, but say they exhibit ringing: now the input to the next gate does not match the output from the proceeding gate. This is why so much effort is spent on maintaining good signal quality.
Explanation:
(1) is necessary, to show that the circuit is not reciprocal. That is, a signal applied to one port does not necessarily come out another port. A passive filter network is reciprocal, i.e., it exhibits the same gain in either direction. An amplifier is usually non-reciprocal (but not necessarily digital). Digital must be non-reciprocal, in order for sequential logic to be implemented. (It's noteworthy that, as an algebra, Boolean algebra needn't be non-reciprocal: indeed, that's one of the special powers of the equals operator. But it's rather difficult to implement and design circuits that are reciprocal. This is an important theoretical aspect of quantum computing, by the way: irreversible logic operations are equivalent to lost information and increased entropy. Quantum computers can only achieve their power by harnessing this behavior!)
(2) allows for more than strict 0/1 Boolean algebra. IEEE 1164 is an example of such a system. In short, the system is extended to allow for outputs driving outputs, as long as they do so in a well-behaved manner. This allows complex bus systems to be built with ease, while allowing the compiler to detect conflicts.
(3) is simply reality: the speed of light is finite, and no physical logic can be truly instantaneous. In analog systems, the lesson again is well learned: there is delay everywhere, and its low-frequency approximations, inductance and capacitance.
Tim