By A. Hartkämper, H. Neumann
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Example text
6 may be slightly varied to give useful results. 4 may be revised as follows: suppose that f ∈ R[a, b] and that there is a continuous function F : [a, b] → R such that F is differentiable b on (a, b) and F ′ (t) = f (t) for all t ∈ (a, b). Then a f = F(b) − F(a). 4. 6 may be similarly revised. 4 is a natural question of existence: which functions in R[a, b] have a primitive? The second fundamental theorem of calculus, given below, provides a partial result in this connection. 4 show that f ∈ R[a, b] has a primitive if, and only if, x the mapping x −→ a f is differentiable and has derivative f .
Define F : J → R by ⎧ x ⎪ if x > a, x ∈ J, ⎨ a f F(x) = 0 if x = a, ⎪ ⎩ a − x f if x < a, x ∈ J. Then F is continuous. If f is right- (left-) continuous at x0 ∈ J , then F is right(left-) differentiable at x0 and ′ ′ F+ (x0 ) = f (x0 ) (F− (x0 ) = f (x0 )). 4 Evaluation of Integrals: Integration and Differentiation 25 In particular, if f is continuous at x0 , then F is differentiable at x 0 and F ′ (x0 ) = f (x0 ). Proof Suppose that b ∈ J , b > a. Then f ∈ R[a, b] and there exists a real number M, depending on b, such that | f (t)| < M if a ≤ t ≤ b.
4) if α = −1. c Since limv→∞ c f exists in R if, and only if, α < −1; and limu→0+ u f exists in R if, and only if, α > −1, (iii) follows immediately. 7. 9 Let a, b ∈ R, a < b; let f : (a, b) → R be bounded and in Rloc (a, b); suppose that g : [a, b] → R is such that g |(a,b) = f . Then g ∈ R[a, b], f is improperly Riemann-integrable over (a, b) and b (I R) a b f = g. 8, g ∈ R[a, b]. 5. 10 Let f : (0, 1) → R be given by f (x) = (log x) log(1 − x) for 0 < x < 1. We claim that f is improperly Riemann-integrable over (0, 1).