Discontinuities in Maggots

Michael Berry's old Physics Today article on singular limits opens with this brilliant analogy:

Biting into an apple and finding a maggot is bad enough, but finding half a maggot is worse. Discovering one-third of a maggot would be more distressing still: The less you find, the more you must have eaten. Extrapolating to the limit, an encounter with no maggot at all should be the ultimate bad-apple experience. This remorseless logic fails, however, because the limit is singular: A very small maggot fraction (f approaching 0) is qualitatively different from no maggot (f = 0). Limits in physics can be singular too [...]