[2] It is relatively rare for such sufficient conditions to be also necessary, so that a sharper piece of analysis may extend the domain of validity of formal results.
Professionally speaking, therefore, analysts push the envelope of techniques, and expand the meaning of well-behaved for a given context.
G. H. Hardy wrote that "The problem of deciding whether two given limit operations are commutative is one of the most important in mathematics".
[3] An opinion apparently not in favour of the piece-wise approach, but of leaving analysis at the level of heuristic, was that of Richard Courant.
Examples abound, one of the simplest being that for a double sequence am,n: it is not necessarily the case that the operations of taking the limits as m → ∞ and as n → ∞ can be freely interchanged.