1.6. Differentiation theorems 135 argument does not particularly care that there is a loss of a constant factor in this estimate. For piecewise constant jump functions, the claim is clear (indeed, the derivative exists and is zero outside of finitely many discontinuities). Now we run the density argument. Let F be a bounded jump function, and let ε 0 and λ 0 be arbitrary. As every jump function is the uniform limit of piecewise constant jump functions, we can find a piecewise constant jump function Fε such that |F (x) − Fε(x)| ≤ ε for all x. Indeed, by taking Fε to be a partial sum of the basic jump functions that make up F , we can ensure that F − Fε is also a monotone non-decreasing function. Applying the second part of Lemma 1.6.26, we have {x ∈ R : D+(F − Fε)(x) ≥ λ} ≤ 2Cε λ for some absolute constant C, and similarly for the other four Dini deriva- tives. Thus, outside of a set of measure at most 8Cε/λ, all of the Dini derivatives of F − Fε are less than λ. Since Fε is almost everywhere dif- ferentiable, we conclude that outside of a set of measure at most 8Cε/λ, all the Dini derivatives of F (x) lie within λ of Fε(x), and in particular, are finite and lie within 2λ of each other. Sending ε to zero (holding λ fixed), we conclude that for almost every x, the Dini derivatives of F are finite and lie within 2λ of each other. If we then send λ to zero, we see that for almost every x, the Dini derivatives of F agree with each other and are finite, and the claim follows. This concludes the proof of Theorem 1.6.25. Just as the integration theory of unsigned functions can be used to de- velop the integration theory of the absolutely convergent functions (see Sec- tion 1.3.4), the differentiation theory of monotone functions can be used to develop a parallel differentiation theory for the class of functions of bounded variation: Definition 1.6.33 (Bounded variation). Let F : R → R be a function. The total variation F TV (R) (or F TV for short) of F is defined to be the supremum F TV (R) := sup x0...xn n i=1 |F (xi) − F (xi+1)| where the supremum ranges over all finite increasing sequences x0,...,xn of real numbers with n ≥ 0 this is a quantity in [0, +∞]. We say that F has bounded variation (on R) if F TV (R) is finite. (In this case, F TV (R) is often written as F BV (R) or just F BV .)

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