1.6. Differentiation theorems 111 Theorem 1.6.9 (First fundamental theorem of calculus). Let [a, b] be a compact interval of positive length. Let f : [a, b] C be a continuous func- tion, and let F : [a, b] C be the indefinite integral F (x) := x a f(t) dt. Then F is differentiable on [a, b], with derivative F (x) = f(x) for all x [a, b]. In particular, F is continuously differentiable. Proof. It suffices to show that lim h→0+ F (x + h) F (x) h = f(x) for all x [a, b), and lim h→0− F (x + h) F (x) h = f(x) for all x (a, b]. After a change of variables, we can write F (x + h) F (x) h = 1 0 f(x + ht) dt for any x [a, b) and any sufficiently small h 0, or any x (a, b] and any sufficiently small h 0. As f is continuous, the function t f(x + ht) converges uniformly to f(x) on [0, 1] as h 0 (keeping x fixed). As the interval [0, 1] is bounded, 1 0 f(x+ht) dt thus converges to 1 0 f(x) dt = f(x), and the claim follows. Corollary 1.6.10 (Differentiation theorem for continuous functions). Let f : [a, b] C be a continuous function on a compact interval. Then we have lim h→0+ 1 h [x,x+h] f(t) dt = f(x) for all x [a, b), lim h→0+ 1 h [x−h,x] f(t) dt = f(x) for all x (a, b], and thus lim h→0+ 1 2h [x−h,x+h] f(t) dt = f(x) for all x (a, b). In these notes we explore the question of the extent to which these the- orems continue to hold when the differentiability or integrability conditions on the various functions F, F , f are relaxed. Among the results proven in this section are: (i) The Lebesgue differentiation theorem, which, roughly speaking, as- serts that Corollary 1.6.10 continues to hold for almost every x if f is merely absolutely integrable, rather than continuous.
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