138 1. Measure theory Exercise 1.6.42 (Lipschitz differentiation theorem, one-dimensional case). A function f : R R is said to be Lipschitz continuous if there exists a constant C 0 such that |f(x) f(y)| C|x y| for all x, y R the smallest C with this property is known as the Lipschitz constant of f. Show that every Lipschitz continuous function F is locally of bounded variation, and hence differentiable almost everywhere. Furthermore, show that the derivative F , when it exists, is bounded in magnitude by the Lipschitz constant of F . Remark 1.6.36. The same result is true in higher dimensions, and is known as the Rademacher differentiation theorem, but we will defer the proof of this theorem to Section 2.2, when we have the powerful tool of the Fubini-Tonelli theorem (Corollary 1.7.23) available, that is particularly useful for deducing higher-dimensional results in analysis from lower-dimensional ones. Exercise 1.6.43. A function f : R R is said to be convex if one has f((1 t)x + ty) (1 t)f(x)+ tf(y) for all x y and 0 t 1. Show that if f is convex, then it is continuous and almost everywhere differentiable, and its derivative f is equal almost everywhere to a monotone non-decreasing function, and so is itself almost everywhere differentiable. (Hint: Drawing the graph of f, together with a number of chords and tangent lines, is likely to be very helpful in providing visual intuition.) Thus we see that in some sense, convex functions are “almost everywhere twice differentiable”. Similar claims also hold for concave functions, of course. 1.6.4. The second fundamental theorem of calculus. We are now fi- nally ready to attack the second fundamental theorem of calculus in the cases where F is not assumed to be continuously differentiable. We begin with the case when F : [a, b] R is monotone non-decreasing. From Theo- rem 1.6.25 (extending F to the rest of the real line if needed), this implies that F is differentiable almost everywhere in [a, b], so F is defined a.e. from monotonicity we see that F is non-negative whenever it is defined. Also, an easy modification of Exercise 1.6.1 shows that F is measurable. One half of the second fundamental theorem is easy: Proposition 1.6.37 (Upper bound for second fundamental theorem). Let F : [a, b] R be monotone non-decreasing (so that, as discussed above, F is defined almost everywhere, is unsigned, and is measurable). Then [a,b] F (x) dx F (b) F (a). In particular, F is absolutely integrable. Proof. It is convenient to extend F to all of R by declaring F (x) := F (b) for x b and F (x) := F (a) for x a, then F is now a bounded monotone
Previous Page Next Page