Assume a case of perfect price inelasticity of demand.
In such a case,
PED = 0
The quantity demanded does not respond to changes in price, i.e. it remains constant, even when price increases or decreases (instead of falling or rising, as it should if it conformed to the inverse relation defined by the law of demand).
Thus, shouldn't perfect price inelasticity also be considered as an exception to the law of demand, just as Giffen goods and Veblen goods are?
Because although some goods have perfect inelasticity over some interval, they are not perfectly inelastic over the whole demand curve - over the rest of the curve, quantity and demand will be strictly inversely related. And the law of demand doesn't have a "strictly" in there. Price and quantity are inversely related, but not strictly so, which (in the mathematical sense) means that parts of the demand curve are allowed to be perfectly inelastic.
It is a bit of an exaggeration to call anything a 'law' in economics. So, there is in the first place not really any 'law of demand' (except in introductory economics). And as such no one is too interested in carefully specifying all the exceptions to the 'law of demand'.
(There is though a 'compensated law of demand', which is a much less sweeping statement than the 'law of demand' - see e.g. Ch 2F of Mas-Colell, Whinston, and Green.)
To use an analogy, the law of gravity says (inter alia) that two bodies attract each other. The 'law' of demand says that the higher price is, the lower the quantity demanded.
But the first law is meant to be an iron-clad rule (possibly subject to corrections later on, as new discoveries are made). Two bodies ALWAYS attract each other. If ever it is found that two bodies do not attract each other, then we must find some way to explain it, or otherwise rethink our understanding of physics.
Whereas the second law is meant as a loose generalization: It tends generally to be the case that higher price ⇒ lower demand. But we are not terribly bothered when this general rule occasionally fails.
If a consumer's demand for a good is perfectly inelastic with respect to the price, this means that the consumer is prepared to spend all his available income to that one good, even if this means that he won't consume anything else. This means that the consumer has lexicographic preferences, with this one good above everything else.
Lexicographic preferences are not continuous, and so they cannot be represented by a continuous utility function. They have indifference "curves" that are single points, etc.
So the case of perfectly inelastic demand represents an "exception" at a deeper level than for example Giffen goods represent. Giffen goods are manageable in the context of "usual" preferences, tweaked but not scrapped altogether.
Finally, note that "perfectly inelastic demand" represented as a straight vertical line, does not answer the question "what will happen if, given his budget constraint, the consumer cannot afford to buy the specific quantity $\bar q$?" Will he buy as much as his budget $B$ allows, or will he buy nothing?
If, when he cannot afford the desired quantity $\bar q$, he buys as much as he can, then in reality the demand curve becomes negatively sloped after a price level, and it acquires a "forced" elasticity, dictated by the budget constraint:
The above represents that case of truly lexicographic preferences. In the case where, if the consumer cannot afford $\bar q$, then he buys no quantity of the good, things are more complicated.
