1.6. Differentiation theorems 133 i.e., a function of the form F = n cnJn, where n ranges over an at most countable set, each Jn is a basic jump function, and the cn are positive reals with n cn ∞. If there are only finitely many n involved, we say that F is a piecewise constant jump function. Thus, for instance, if q1,q2,q3,... is any enumeration of the rationals, then ∑∞ n=1 2−n1[qn,+∞) is a jump function. Clearly, all jump functions are monotone non-decreasing. From the ab- solute convergence of the cn we see that every jump function is the uniform limit of piecewise constant jump functions, for instance, ∑∞ n=1 cnJn is the uniform limit of ∑N n=1 cnJn. One consequence of this is that the points of discontinuity of a jump function ∑∞ n=1 cnJn are precisely those of the individual summands cnJn, i.e., of the points xn where each Jn jumps. The key fact is that these functions, together with the continuous mono- tone functions, essentially generate all monotone functions, at least in the bounded case: Lemma 1.6.31 (Continuous-singular decomposition for monotone func- tions). Let F : R R be a monotone non-decreasing function. (i) The only discontinuities of F are jump discontinuities. More pre- cisely, if x is a point where F is discontinuous, then the limits limy→x− F (y) and limy→x+ F (y) both exist, but are unequal, with limy→x− F (y) limy→x+ F (y). (ii) There are at most countably many discontinuities of F . (iii) If F is bounded, then F can be expressed as the sum of a continuous monotone non-decreasing function Fc and a jump function Fpp. Remark 1.6.32. This decomposition is part of the more general Lebesgue decomposition, discussed in §1.2 of An epsilon of room, Vol. I. Proof. By monotonicity, the limits F−(x) := limy→x− F (y) and F +(x) := limy→x+ F (y) always exist, with F−(x) F (x) F+(x) for all x. This gives (i). By (i), whenever there is a discontinuity x of F , there is at least one rational number qx strictly between F−(x) and F+(x), and from monotonic- ity, each rational number can be assigned to at most one discontinuity. This gives (ii). Now we prove (iii). Let A be the set of discontinuities of F , thus A is at most countable. For each x A, we define the jump cx := F+(x) F−(x) 0, and the fraction θx := F (x)−F−(x) F+(x)−F−(x) [0, 1]. Thus F+(x) = F−(x) + cx and F (x) = F−(x) + θxcx.
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