A Second Course on Real Functions

Abstract

When considering a mathematical theorem one ought not only to know how to prove it but also why and whether any given conditions are necessary. All too often little attention is paid to to this side of the theory and in writing this account of the theory of real functions the authors hope to rectify matters. They have put the classical theory of real functions in a modern setting and in so doing have made the mathematical reasoning rigorous and explored the theory in much greater depth than is customary. The subject matter is essentially the same as that of ordinary calculus course and the techniques used are elementary (no topology, measure theory or functional analysis). Thus anyone who is acquainted with elementary calculus and wishes to deepen their knowledge should read this.

Simultaneous monotone approximation in low-order mean

“…Example (19) in Section 4 shows that not all members of fii(f,g; M) preserve the continuity of / and g. As a consequence of (3) and (13)…”

“…Since / and g have at most discontinuities of the first kind, they can be uniformly approximated by step functions (see [19]). Thus, for any n G N there are step functions t=2 and i=2 (where XA is the indicator function of A, that is, XA(0 = 1 if < G A and XA(*) = 0 ii t $ A) such that \\f -f 71^ < n " 1 and \\g -g 71^ < n " 1 , where {0 = t 0 < h < ... < t n = 1} is the common refinement of the partitions of [0,1] associated with the canonical representations of f n and g n .…”

Lineability in sequence and function spaces

“…In view of the last theorem one might believe that the expression "f = 0 on a dense set" (see the definition of P) could be replaced by the stronger one "f = 0 almost everywhere". But this is not possible because every differentiable function is an N-function -that is, it sends sets of null measure into sets of null measure-(see [29,Theorem 21.9]) and every continuous N-function on an interval whose derivative vanishes almost everywhere must be a constant (see [29,Theorem 21.10] Since it is not finite dimensional, a simple application of Baire's category theorem yields dim (BDP) = c. Now, on one hand, we have that, trivially, BDP is dense-lineable in itself. On the other hand, it is known that the set of derivatives that are positive on a dense set and negative on another is a dense G δ set in the Banach space BDP [13, p. 34].…”

The spans in Brownian motion

“…It suffices to prove that for rationals 0 ≤ p < q < r < s ≤ 1,M pq ∩ M rs = ∅ a.s.Note that for any function f : R → R the set of levels of local extrema is countable, see e.g. van Rooij and Schikhof[147, Theorem 7.2]. Thus, M pq and M rs are both countable.…”