on a set of measure zero. Note that points of nonzero oscillation (i.e., points at which f is "badly behaved") are discontinuities which, unless they make up a set of zero, are confined to a negligible set. Functions from topological spaces to complete lattices Functions from metric spaces There is a notion of limsup and liminf for functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the limsup, liminf, and the limit of a real sequence. Take a metric space , a subspace contained in , and a function . Define, for any limit point of , and where denotes the metric ball of radius about . Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we have and similarly Functions from topological spaces This finally motivates the definitions for general topological spaces. Take X, E and a as before, but now let X be a topological space. In this case, we replace metric balls with neighborhoods: (there is a way to write the formula using "lim" using nets and the neighborhood filter). This version is often useful in discussions of semi-continuity which crop up in analysis quite often. An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the closure of N in [−∞,∞], the extended real number line, is N ∪ {∞}.) Sequences of sets The power set ℘(X) of a set X is a complete lattice that is ordered by set inclusion, and so the supremum and infimum of any set of subsets (in terms of set inclusion) always exist. In particular, every subset Y of X is bounded above by X and below by the empty set ∅ because ∅ ⊆ Y ⊆ X. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(X) (i.e., sequences of subsets of X). There are two common ways to define the limit of sequences of sets. In both cases: The sequence accumulates around sets of points rather than single points themselves. That is, because each element of the sequence is itself a set, there exist accumulation sets that are somehow nearby to infinitely many elements of the sequence. The supremum/superior/outer limit is