Riemann Integrable Functions

Transcription

1 Riemann Integrable Functions MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007

2 Cauchy Criterion Theorem (Cauchy Criterion) A function f : [a, b] R belongs to R[a, b] if and only if for every ǫ > 0 there exists η ǫ > 0 such that if Ṗ and Q are any tagged partitions of [a, b] with Ṗ < η ǫ and Q < η ǫ, then S(f; Ṗ) S(f; Q) < ǫ.

3 Cauchy Criterion Theorem (Cauchy Criterion) A function f : [a, b] R belongs to R[a, b] if and only if for every ǫ > 0 there exists η ǫ > 0 such that if Ṗ and Q are any tagged partitions of [a, b] with Ṗ < η ǫ and Q < η ǫ, then S(f; Ṗ) S(f; Q) < ǫ.

4 Example Example { 1 2 if 0 x 2, g(x) = 1 if 2 < x 3 { 2 1 if x [0, 1] Q, h(x) = 0 if x R\Q R[0, 3] / R[0, 1]

5 Squeeze Theorem Theorem (Squeeze Theorem) Let f : [a, b] R. Then f R[a, b] if and only if for every ǫ > 0 there exist functions α ǫ and ω ǫ in R[a, b] with α ǫ (x) f(x) ω ǫ (x) for all x [a, b], and such that b a (ω ǫ α ǫ ) < ǫ.

6 Squeeze Theorem Theorem (Squeeze Theorem) Let f : [a, b] R. Then f R[a, b] if and only if for every ǫ > 0 there exist functions α ǫ and ω ǫ in R[a, b] with α ǫ (x) f(x) ω ǫ (x) for all x [a, b], and such that b a (ω ǫ α ǫ ) < ǫ.

7 Step Functions Definition Let I R be an interval and let φ : I R. Then φ is called a step function if it has only a finite number of distinct values, each value being assumed on one or more intervals in I. Lemma If J is a subinterval of [a, b] having endpoints c < d and if φ J (x) = 1 for x J and φ J (x) = 0 elsewhere in [a, b], then φ J R[a, b] and b a φ J = d c.

8 Step Functions Definition Let I R be an interval and let φ : I R. Then φ is called a step function if it has only a finite number of distinct values, each value being assumed on one or more intervals in I. Lemma If J is a subinterval of [a, b] having endpoints c < d and if φ J (x) = 1 for x J and φ J (x) = 0 elsewhere in [a, b], then φ J R[a, b] and b a φ J = d c.

9 Step Functions Definition Let I R be an interval and let φ : I R. Then φ is called a step function if it has only a finite number of distinct values, each value being assumed on one or more intervals in I. Lemma If J is a subinterval of [a, b] having endpoints c < d and if φ J (x) = 1 for x J and φ J (x) = 0 elsewhere in [a, b], then φ J R[a, b] and b a φ J = d c.

10 Step Functions (cont.) Theorem If φ : [a, b] R is a step function, then φ R[a, b].

11 Step Functions (cont.) Theorem If φ : [a, b] R is a step function, then φ R[a, b].

12 Continuous Functions Theorem If f : [a, b] R is a continuous function, then f R[a, b].

13 Continuous Functions Theorem If f : [a, b] R is a continuous function, then f R[a, b].

14 Monotone Functions Theorem If f : [a, b] R is a monotone on [a, b], then f R[a, b].

15 Monotone Functions Theorem If f : [a, b] R is a monotone on [a, b], then f R[a, b].

16 Additivity Theorem Theorem (Additivity Theorem) Let f : [a, b] R and let c (a, b). Then f R[a, b] if and only if its restrictions to [a, c] and [c, b] are both Riemann integrable. In this case b c b f = f + f. c a a

17 Additivity Theorem Theorem (Additivity Theorem) Let f : [a, b] R and let c (a, b). Then f R[a, b] if and only if its restrictions to [a, c] and [c, b] are both Riemann integrable. In this case b c b f = f + f. c a a

18 Related Results Corollary If f R[a, b], and if [c, d] [a, b], then the restriction of f to [c, d] is in R[c, d]. Corollary If f R[a, b], and if a = c 0 < c 1 < < c m = b, then the restrictions of f to each of the subintervals [c i 1, c i ] are Riemann integrable and b a f = m i=1 ci c i 1 f.

19 Related Results Corollary If f R[a, b], and if [c, d] [a, b], then the restriction of f to [c, d] is in R[c, d]. Corollary If f R[a, b], and if a = c 0 < c 1 < < c m = b, then the restrictions of f to each of the subintervals [c i 1, c i ] are Riemann integrable and b a f = m i=1 ci c i 1 f.

20 Related Results Corollary If f R[a, b], and if [c, d] [a, b], then the restriction of f to [c, d] is in R[c, d]. Corollary If f R[a, b], and if a = c 0 < c 1 < < c m = b, then the restrictions of f to each of the subintervals [c i 1, c i ] are Riemann integrable and b a f = m i=1 ci c i 1 f.

21 Exchanging Limits of Integration Definition If f R[a, b] and if α,β [a, b] with α < β, we define α β β f = f α and α α f = 0 Theorem If f R[a, b] and if α,β,γ are any numbers in [a, b], then β α f = γ α β f + f, γ in the sense that the existence of any two of these integrals implies the existence of the third integral and the equality.

22 Exchanging Limits of Integration Definition If f R[a, b] and if α,β [a, b] with α < β, we define α β β f = f α and α α f = 0 Theorem If f R[a, b] and if α,β,γ are any numbers in [a, b], then β α f = γ α β f + f, γ in the sense that the existence of any two of these integrals implies the existence of the third integral and the equality.

23 Exchanging Limits of Integration Definition If f R[a, b] and if α,β [a, b] with α < β, we define α β β f = f α and α α f = 0 Theorem If f R[a, b] and if α,β,γ are any numbers in [a, b], then β α f = γ α β f + f, γ in the sense that the existence of any two of these integrals implies the existence of the third integral and the equality.

24 Homework Read Section 7.2. Pages : 1, 3, 6, 8, 16, 17, 19