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1.6. Differentiation theorems 145 Combining this result with Exercise 1.6.49, we obtain a satisfactory clas- sification of the absolutely continuous functions: Exercise 1.6.51. Show that a function F : [a, b] → R is absolutely con- tinuous if and only if it takes the form F (x) = [a,x] f(y) dy + C for some absolutely integrable f : [a, b] → R and a constant C. Exercise 1.6.52 (Compatibility of the strong and weak derivatives in the absolutely continuous case). Let F : [a, b] → R be an absolutely continu- ous function, and let φ: [a, b] → R be a continuously differentiable func- tion supported in a compact subset of (a, b). Show that [a,b] F φ(x) dx = − [a,b] Fφ (x) dx. Inspecting the proof of Theorem 1.6.40, we see that the absolute conti- nuity was used primarily in two ways: first, to ensure the almost everywhere existence, and then to control an exceptional null set E. It turns out that one can achieve the latter control by making a different hypothesis, namely that the function F is everywhere differentiable rather than merely almost everywhere differentiable. More precisely, we have Proposition 1.6.41 (Second fundamental theorem of calculus, again). Let [a, b] be a compact interval of positive length, let F : [a, b] → R be a differ- entiable function, such that F is absolutely integrable. Then the Lebesgue integral [a,b] F (x) dx of F is equal to F (b) − F (a). Proof. This will be similar to the proof of Theorem 1.6.40, the one main new twist being that we need several open sets U instead of just one. Let E ⊂ [a, b] be the set of points x which are not Lebesgue points of F , together with the endpoints a, b. This is a null set. Let ε 0, and then let κ 0 be a small enough number such that U |F (x)| dx ≤ ε whenever U is measurable with m(U) ≤ κ. We can also ensure that κ ≤ ε. For every natural number m = 1, 2,... we can find an open set Um con- taining E of measure m(Um)≤κ/4m. In particular, we see that m( ∞ m=1 Um) ≤ κ and thus ∞ m=1 Um |F (x)| dx ≤ ε. Now define a gauge function δ : [a, b] → (0, +∞) as follows. (i) If x ∈ E, we define δ(x) 0 to be a small enough number such that the open interval (x − δ(x),x + δ(x)) lies in Um, where m is the first natural number such that |F (x)| ≤ 2m, and also small enough such that |F (y) − F (x) − (y − x)F (x)| ≤ ε|y − x| holds whenever |y − x| ≤ δ(x). (Here we crucially use the everywhere differentiability to ensure that f (x) exists and is finite here.)