1.6. Differentiation theorems 137 F + : R → R of F by the formula (1.30) F +(x) := sup x0...xn≤x n i=1 max(F (xi+1) − F (xi), 0). It is clear from construction that this is a monotone increasing function, taking values between 0 and F TV (R), and is thus bounded. To conclude the proposition, it suﬃces to (by writing F = F+ − (F+ − F−) to show that F+ − F is non-decreasing, or in other words, to show that F +(b) ≥ F +(a) + F (b) − F (a). If F (b)−F (a) is negative, then this is clear from the monotone non-decreasing nature of F +, so assume that F (b) − F (a) ≥ 0. But then the claim fol- lows because any sequence of real numbers x0 . . . xn ≤ a can be extended by one or two elements by adding a and b, thus increasing the sum supx0...xn ∑n i=1 max(F (xi) − F (xi+1), 0) by at least F (b) − F (a). Exercise 1.6.40. Let F : R → R be of bounded variation. Define the positive variation F + by (1.30), and the negative variation F − by F −(x) := sup x0...xn≤x n i=1 max(−F (xi+1) + F (xi), 0). Establish the identities F (x) = F (−∞) + F +(x) − F −(x), F TV [a,b] = F +(b) − F +(a) + F −(b) − F −(a), and F TV = F +(+∞) + F −(+∞) for every interval [a, b], where F (−∞) := limx→−∞ F (x), F +(+∞) := limx→+∞ F +(x), and F −(+∞) := limx→+∞ F −(x). (Hint: The main diﬃ- culty comes from the fact that a partition x0 . . . xn ≤ x that is good for F + need not be good for F −, and vice versa. However, this can be fixed by taking a good partition for F + and a good partition for F − and combining them together into a common refinement.) From Proposition 1.6.34 and Theorem 1.6.25 we immediately obtain Corollary 1.6.35 (BV differentiation theorem). Every bounded variation function is differentiable almost everywhere. Exercise 1.6.41. Call a function locally of bounded variation if it is of bounded variation on every compact interval [a, b]. Show that every function that is locally of bounded variation is differentiable almost everywhere.

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