This paper is related to two others by the same authors, Research Reports LSE-CDAM-2007-23 and 24. In the first, we make a systematic study of properties that hold `automatically', for example, automatic continuity. This topic has been extensively studied for Banach algebras, but it has roots in classical real analysis, and that is the setting here. The second paper is on `analytic automaticity', where the test set on which a weaker property is to imply a stronger one is analytic, and `thin'. The roots of the present paper lie in an important (but little cited) paper of Kestelman [Kes], and a subsequent paper of Borwein and Ditor [BoDi]. There the setting is measure-theoretic, and one assumes sets measurable, but one can also proceed topologically, and assume sets to have the Baire property (to be `Baire'). In both contexts one has the sequence containment property at generically all points (that is, avoiding a `thin' exceptional set -- which may be null or meagre): typically, a subsequence of deviations from a location in the set, fixed according to a given sequence of deviations, stays within the set. Instead of deviations in this pointwise mode, as in [Kes], [BoDi] and Research Report LSE-CDAM-2007-23, here we consider functional deviations, again obtaining generic conclusions; these permit a form of fixed-point property to hold
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