# Piecewise Continuous φ

Size: px
Start display at page:

Transcription

1 Piecewise Continuous φ φ is piecewise continuous on [, b] if nd only if b in R nd φ : [, b] C There is finite set S [, b] such tht, for ll t [, b] S, φ is continuous t t: φ(t) = lim φ(u) u t u [,b] For ll t [, b), φ hs finite right limit t t. For ll t (, b], φ hs finite left limit t t. Rmsey Complex Integrtion

2 Piecewise Continuous, Slide 2 Lemm: φ is piecewise continuous on [, b] if nd only if R(φ) nd I(φ) re piecewise continuous on [, b]. Theorem from Advnced Clculus: f : [, b] R is Riemnn integrble if nd only if f is bounded nd the set of discontinuities of f hs outer mesure 0. Specil Problem (Chi): Let f be rel-vlued nd piecewise continuous on [, b]. Then 1 f is bounded 2 f is Riemnn integrble Rmsey Complex Integrtion

3 Piecewise Continuous, Slide 3 Specil Problem (Kodgis): Assume tht φ is piecewise continuous on [, b]. g : E C with E C g is continuous on n open set U E There is closed set F such tht rnge(φ) F U. Then g φ is piecewise continuous on [, b]. Exmple: Note tht the bsolute vlue function is continuous on ll of C. Consequently, if φ is piecewise continuous, so is φ. This is used implicitly in Section 3 of Chpter 6 on pge 66. Rmsey Complex Integrtion

4 Piecewise Continuous, Slide 4 Specil Problem (Chevlier): Suppose tht φ is piecewise continuous on [, b]. Suppose tht τ : [c, d] [, b] is one-to-one nd continuous. Then φ τ is piecewise continuous on [c, d]. Hint: You cn prove tht τ is strictly incresing or strictly decresing. Tht leds to two cses. Rmsey Complex Integrtion

5 Integrls of piecewise continuous φ Def n: If φ is piecewise continuous on [, b] we define φ(t) dt by φ(t) dt := R(φ(t)) dt + i I(φ(t)) dt Specil Problem (Reckwerdt): Let φ nd λ be piecewise continuous on [, b]. Then φ + λ is piecewise continuous on [, b], nd (φ + λ)(t) dt = φ(t) dt + λ(t) dt Rmsey Complex Integrtion

6 Integrls of piecewise continuous φ, Slide 2 Specil Problem (Felicino): If φ is piecewise continuous on [, b] nd c is complex number, then cφ is piecewise continuous on [, b] nd (cφ)(t) dt = c φ(t) dt Theorem, pge 65: If φ 1 nd φ 2 re piecewise continuous on [, b], nd c 1 nd c 2 re complex numbers, then c 1 φ 1 + c 2 φ 2 is piecewise continuous on [, b] nd [c 1 φ 1 + c 2 φ 2 ](t) dt = c 1 φ 1 (t) dt + c 2 φ 2 (t) dt Rmsey Complex Integrtion

7 Integrls of piecewise continuous φ, Slide 3 Specil Problem (Rder) Suppose tht φ is piecewise continuous on [, b]. Suppose tht [c, d] [, b] with c < d. Then d d δ φ(t) dt = lim φ(t) dt c δ 0 c+δ Proof Hints: You my use for free tht, when 0 < δ < (d c)/2, c+δ c d δ d d φ(t) dt + φ(t) dt + φ(t) dt = φ(t) dt c+δ d δ c Somehow use the bounded-ness of φ to finish the rgument. Rmsey Complex Integrtion

8 Integrls of piecewise continuous φ, Silde 4 The previous problem is vlid more generlly. It remins vlid if R(φ) nd I(φ) re Riemnn integrble on [, b] R(φ) nd I(φ) re Lebesgue mesurble on [, b] with R(φ(t)) dt < nd I(φ(t)) dt < Rmsey Complex Integrtion

9 Integrls of piecewise continuous φ, Slide 5 Specil Problem (Hedley): Let φ nd ψ be piecewise continuous on [, b]. Suppose there is finite set W such tht, for ll t [, b] W, φ(t) = ψ(t) Then φ(t) dt = ψ(t) dt Rmsey Complex Integrtion

10 Integrls of piecewise continuous φ, Slide 6 Lemm: Let φ nd ψ be piecewise continuous on [, b]. Then φ ψ is piecewise continuous on [, b]. Lemm: Let φ : [, b] C. Let < c < b. Set λ equl to the restriction of φ to [, c] nd ψ equl to the restriction of φ to [c, b]. Then φ is piecewise continuous on [, b] if nd only if λ is piecewise continuous on [, c] nd ψ is piecewise continuous on [c, b]. If φ is piecewise continuous on [, b], then φ(t) dt = c λ(t) dt + ψ(t) dt c Rmsey Complex Integrtion

11 Differentiting φ : [, b] C Def n: Let < b nd φ : [, b] C. We sy tht φ is differentible t t 0 if nd only if t 0 [, b] nd there is complex number L such tht φ(u) φ(t 0 ) L = lim u t0 u t 0 u [,b] L is necessrily unique, nd we cll it φ (t 0 ). Specil Problem (Billington): Let < b nd φ : [, b] C. Then 1 φ is differentible t t 0 if nd only if R(φ) nd I(φ) re differentible t t 0. 2 If φ is differentible t t 0 then φ (t 0 ) = [R(φ)] (t 0 ) + i [I(φ)] (t 0 ) Note tht this problem sys tht, when φ (t) exists, [R(φ)] (t) = R[φ (t)] nd [I(φ)] (t) = I[φ (t)]. Rmsey Complex Integrtion

12 Piecewise C 1 Def n: Let < b nd φ : [, b] C. φ is piecewise C 1 if nd only if φ is continuous on [, b] There is finite set S such tht, for ll t [, b] S φ is differentible t t. For ll t [, b], if φ is differentible t t then φ is continuous t t. For ll t [, b), φ hs finite right limit t t. For ll t (, b], φ hs finite left limit t t. Rmsey Complex Integrtion

13 Piecewise C 1, Slide 2 Specil Problem (Holmes): Let < b, φ : [, b] : C, nd φ piecewise C 1. Suppose tht g : E C, with E C. Let F G E such tht F is closed nd G is open The rnge of φ is subset of F g is holomorphic on G g is C 1 on G (mening, if g = u + iv with u nd v rel, u x (z), u y (z), v x (z) nd v y (z) re continuous t every z G). Then g φ is piecewise C 1. Rmsey Complex Integrtion

14 Piecewise C 1, Slide 3 Specil Problem (Thompson): Let < b, φ : [, b] C, nd φ piecewise C 1. Let τ : [c, d] [, b] be one-to-one with continuous derivtive (C 1 ). Then φ τ is piecewise C 1. Hint: Prove tht τ must be strictly decresing or strictly incresing. Tht leds to two cses. Rmsey Complex Integrtion

15 Piecewise C 1, Slide 4 Lemm: Let < b, φ : [, b] C. Then φ is piecewise C 1 if nd only if R(φ) nd I(φ) re piecewise C 1. Fundmentl Theorem of Clculus: Let < b, φ : [, b] C. Suppose tht φ is piecewise C 1. Then S is finite, where S is the set of t [, b] such tht φ is not differentible t t. Let h : [, b] C stisfy h(t) = φ (t) for t [, b] S. h on S cn be n rbitrry, independent ssignment of complex numbers. Then h is Riemnn integrble nd h(t) dt = φ(b) φ() Rmsey Complex Integrtion

16 Proof of the Fundmentl Theorem of Clculus Clim 1: h is continuous t t [, b] S. Proof: Let t / S. Becuse S is finite, there is some δ 0 > 0 such tht u t < δ 0 u / S Thus, if u t < δ 0 nd u [, b], we hve h(u) = φ (u). Becuse φ (t) exists (since t / S), φ is continuous t t by hypothesis. Becuse φ nd h gree on [, b] (t δ 0, t + δ 0 ) we hve h continuous t t. Rmsey Complex Integrtion

17 Proof Continued, Slide 2 Clim 2: For ll t [, b), h hs finite right limit t t. Proof: Let t [, b). If t / S, then h is continuous t t. Becuse t = b, this implies tht the right limit of h t t is h(t), which is finite. Suppose tht t S. Becuse S is finite, there is some δ 0 > 0 such tht (t δ 0, t + δ 0 ) S = {t}. Hence u (t, min{t + δ 0, b}) h(u) = φ (u) Note tht both t + δ 0 nd b re strictly bigger thn t. Thus h nd φ gree on n open intervl of positive length with left end t. Becuse φ hs finite right limit t t, so does h. Rmsey Complex Integrtion

18 Proof Continued, Slide 3 Clim 3: For ll t (, b], h hs finite left limit t t. Proof: Let t [, b). If t / S, then h is continuous t t. Becuse t =, this implies tht the left limit of h t t is h(t), which is finite. Suppose tht t S. Becuse S is finite, there is some δ 0 > 0 such tht (t δ 0, t + δ 0 ) S = {t}. Hence u (min{, t δ 0 }, t) h(u) = φ (u) Note tht both t δ 0 nd re strictly less thn t. Thus h nd φ gree on n open intervl of positive length with right end t. Becuse φ hs finite left limit t t, so does h. Rmsey Complex Integrtion

19 Proof Continued, Slide 4 By Clims 1, 2 nd 3, h is piecewise continuous on [, b] nd thus Riemnn integrble. Clim 4: Let c < d in [, b] with (c, d) S =. Then d c h(t) dt = φ(d) φ(c) Proof: Consider δ < (d c)/2. Then J = [c + δ, d δ] (c, d). Hence no point of S is in J nd thus h(u) = φ (u) for u J. Note tht φ is continuous on J becuse it is continuous on the open intervl (c, d) contining J. Becuse φ = [R(φ)] + i[i(φ)], tht mkes both [R(φ)] nd [I(φ)] continuous on J. Rmsey Complex Integrtion

20 Proof Continued, Slide 5 By the Fundmentl Theorem of Clculus, pplied to the rel nd imginry prts of φ, we hve d δ c+δ h(t) dt = = = d δ c+δ d δ c+δ d δ c+δ φ (t) dt R[φ (t)] dt + i [R(φ)] (t) dt + i d δ c+δ d δ c+δ I[φ (t)] dt = {[Re(φ)](d δ) [R(φ)](c + δ)} [I(φ)] (t) dt + i {[I(φ)](d δ) [I(φ)](c + δ)} = φ(d δ) φ(c + δ) Rmsey Complex Integrtion

21 Proof Continued, Slide 6 By Rder s Specil Problem, Therefore d c d c d δ h(t) dt = lim h(t) dt δ 0 c+δ h(t) dt = lim δ 0 {φ(d δ) φ(c + δ)} Becuse φ is C 1, φ is continuous. Therefore the left limit t d of φ is φ(d) nd the right limit t c of φ is φ(c). Hence, since difference in C preserves limits, d c h(t) dt = φ(d) φ(c) Rmsey Complex Integrtion

22 Proof Continued, Slide 7 Suppose S (, b) =. Apply Clim 4 with [c, d] = [, b] to conclude tht h(t) dt = φ(b) φ() Suppose tht S (, b) =. Let {s j } n j=1 enumerte S (, b), with s j < s j+1 for 1 j n 1. Let s 0 = nd s n+1 = b. For 0 j n, (s j, s j+1 ) S =. By Clim 4, sj+1 s j h(t) dt = φ(s j+1 ) φ(s j ) Rmsey Complex Integrtion

23 Proof Continued, Slide 8 Hence h(t) dt = = n j=0 sj+1 s j h(t) dt n [ φ(sj+1 ) φ(s j ) ] j=0 = φ(s n+1 ) φ(s 0 ) = φ(b) φ() Rmsey Complex Integrtion

24 Tringle Inequlity for Integrtion Theorem: If φ is piecewise continuous on [, b], then φ(t) dt φ(t) dt Note: By one of the specil problems, φ is piecewise continuous on [, b] nd hence Riemnn integrble. Rmsey Complex Integrtion

25 Arc Length (Totl Distnce Trveled) Lemm: Let < b, γ : [, b] C, nd γ be piecewise C 1. Suppose tht h 1 : [, b] C nd h 2 : [, b] C such tht, for t [, b] with γ differentible t t, h 1 (t) = h 2 (t) = γ (t) Then h 1 nd h 2 re piecewise continuous on [, b] h 1 nd h 2 re Riemnn integrble nd h 1 (t) dt = h 2 (t) dt Proof: As in the just-completed proof of complex version of the FTC, both h 1 nd h 2 re piecewise continuous. By specil problem, h 1 nd h 2 re piecewise continuous. By nother specil problem, both re Riemnn integrble. Rmsey Complex Integrtion

26 Proof Continued, Slide 2 Let S be the set of t [, b] t which γ is not differentible. Clim 1: Suppose tht c < d with [c, d] [, b] nd (c, d) S =. Then d d h 1 (t) dt = h 2 (t) dt c c Proof: On (c, d), h j (t) = γ (t) = h 2 (t). Hence, for 0 < δ < (d c)/2, for j = 1 nd j = 2 we hve d δ c+δ h j (t) dt = Then for j = 1 nd j = 2 we hve d c h j (t) dt = lim δ 0 d δ c+δ d δ c+δ γ (t) dt γ (t) dt Rmsey Complex Integrtion

27 Proof Continued, Slide 3 Suppose tht S (, b) =. Let [c, d] = [, b] in Clim 1 to conclude tht h 1 (t) dt = h 2 (t) dt Now suppose tht S (, b) =. Enumerte S (, b) s {s j } n j=1. Let = s 0 nd b = s n+1. For 0 j n, (s j, s j+1 ) S =. By Clim 1, sj+1 s j h 1 (t) dt = sj+1 s j h 2 (t) dt Rmsey Complex Integrtion

28 Proof Concluded, Slide 4 Finlly, h 1 (t) dt = = = n sj+1 j=0 s j sj+1 n j=0 s j h 2 (t) dt h 1 (t) dt h 2 (t) dt Rmsey Complex Integrtion

29 The Definition of Arc Length (s Totl Distnce Trveled) Def n: Let < b nd γ : [, b] C be peicewise C 1. Let h : [, b] C such tht, for ll t [, b], γ is differentible t t, then h(t) = γ (t). Then the length of γ is L(γ) = h(t) dt In the book, Srson writes simply γ (t) dt. Rmsey Complex Integrtion

### 7.2 Riemann Integrable Functions

7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous

### W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying

Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)

### Problem Set 4: Solutions Math 201A: Fall 2016

Problem Set 4: s Mth 20A: Fll 206 Problem. Let f : X Y be one-to-one, onto mp between metric spces X, Y. () If f is continuous nd X is compct, prove tht f is homeomorphism. Does this result remin true

### Chapter 4. Lebesgue Integration

4.2. Lebesgue Integrtion 1 Chpter 4. Lebesgue Integrtion Section 4.2. Lebesgue Integrtion Note. Simple functions ply the sme role to Lebesgue integrls s step functions ply to Riemnn integrtion. Definition.

### The Regulated and Riemann Integrals

Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous

### Math 554 Integration

Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

### NOTES AND PROBLEMS: INTEGRATION THEORY

NOTES AND PROBLEMS: INTEGRATION THEORY SAMEER CHAVAN Abstrct. These re the lecture notes prepred for prticipnts of AFS-I to be conducted t Kumun University, Almor from 1st to 27th December, 2014. Contents

### Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when

### A PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS USING HAUSDORFF MEASURES

INROADS Rel Anlysis Exchnge Vol. 26(1), 2000/2001, pp. 381 390 Constntin Volintiru, Deprtment of Mthemtics, University of Buchrest, Buchrest, Romni. e-mil: cosv@mt.cs.unibuc.ro A PROOF OF THE FUNDAMENTAL

### Calculus in R. Chapter Di erentiation

Chpter 3 Clculus in R 3.1 Di erentition Definition 3.1. Suppose U R is open. A function f : U! R is di erentible t x 2 U if there exists number m such tht lim y!0 pple f(x + y) f(x) my y =0. If f is di

### INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

### Chapter 6. Riemann Integral

Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl

### Advanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015

Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n

### II. Integration and Cauchy s Theorem

MTH6111 Complex Anlysis 2009-10 Lecture Notes c Shun Bullett QMUL 2009 II. Integrtion nd Cuchy s Theorem 1. Pths nd integrtion Wrning Different uthors hve different definitions for terms like pth nd curve.

### The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

### Principles of Real Analysis I Fall VI. Riemann Integration

21-355 Principles of Rel Anlysis I Fll 2004 A. Definitions VI. Riemnn Integrtion Let, b R with < b be given. By prtition of [, b] we men finite set P [, b] with, b P. The set of ll prtitions of [, b] will

### Notes on length and conformal metrics

Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### Phil Wertheimer UMD Math Qualifying Exam Solutions Analysis - January, 2015

Problem 1 Let m denote the Lebesgue mesure restricted to the compct intervl [, b]. () Prove tht function f defined on the compct intervl [, b] is Lipschitz if nd only if there is constct c nd function

### Presentation Problems 5

Presenttion Problems 5 21-355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).

### UNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3

UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,

### 4.4 Areas, Integrals and Antiderivatives

. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

### 11 An introduction to Riemann Integration

11 An introduction to Riemnn Integrtion The PROOFS of the stndrd lemms nd theorems concerning the Riemnn Integrl re NEB, nd you will not be sked to reproduce proofs of these in full in the exmintion in

### MATH , Calculus 2, Fall 2018

MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly

### f(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all

3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the

### arxiv: v1 [math.ca] 11 Jul 2011

rxiv:1107.1996v1 [mth.ca] 11 Jul 2011 Existence nd computtion of Riemnn Stieltjes integrls through Riemnn integrls July, 2011 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde

### The Henstock-Kurzweil integral

fculteit Wiskunde en Ntuurwetenschppen The Henstock-Kurzweil integrl Bchelorthesis Mthemtics June 2014 Student: E. vn Dijk First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. A. vn der Schft

### INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable

INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd

### A BRIEF INTRODUCTION TO UNIFORM CONVERGENCE. In the study of Fourier series, several questions arise naturally, such as: c n e int

A BRIEF INTRODUCTION TO UNIFORM CONVERGENCE HANS RINGSTRÖM. Questions nd exmples In the study of Fourier series, severl questions rise nturlly, such s: () (2) re there conditions on c n, n Z, which ensure

### Mapping the delta function and other Radon measures

Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

### SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL (1 + µ(f n )) f(x) =. But we don t need the exact bound.) Set

SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL 28 Nottion: N {, 2, 3,...}. (Tht is, N.. Let (X, M be mesurble spce with σ-finite positive mesure µ. Prove tht there is finite positive mesure ν on (X, M such

### Lecture 1: Introduction to integration theory and bounded variation

Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You

### Review of Riemann Integral

1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.

### Review. April 12, Definition 1.2 (Closed Set). A set S is closed if it contains all of its limit points. S := S S

Review April 12, 2017 1 Definitions nd Some Theorems 1.1 Topology Definition 1.1 (Limit Point). Let S R nd x R. Then x is limit point of S if, for ll ɛ > 0, V ɛ (x) = (x ɛ, x + ɛ) contins n element s S

### Week 10: Riemann integral and its properties

Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the

### Math 118: Honours Calculus II Winter, 2005 List of Theorems. L(P, f) U(Q, f). f exists for each ǫ > 0 there exists a partition P of [a, b] such that

Mth 118: Honours Clculus II Winter, 2005 List of Theorems Lemm 5.1 (Prtition Refinement): If P nd Q re prtitions of [, b] such tht Q P, then L(P, f) L(Q, f) U(Q, f) U(P, f). Lemm 5.2 (Upper Sums Bound

### Chapter 22. The Fundamental Theorem of Calculus

Version of 24.2.4 Chpter 22 The Fundmentl Theorem of Clculus In this chpter I ddress one of the most importnt properties of the Lebesgue integrl. Given n integrble function f : [,b] R, we cn form its indefinite

### This is a short summary of Lebesgue integration theory, which will be used in the course.

3 Chpter 0 ntegrtion theory This is short summry of Lebesgue integrtion theory, which will be used in the course. Fct 0.1. Some subsets (= delmängder E R = (, re mesurble (= mätbr in the Lebesgue sense,

### n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1

The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the

### Calculus I-II Review Sheet

Clculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing

### The Banach algebra of functions of bounded variation and the pointwise Helly selection theorem

The Bnch lgebr of functions of bounded vrition nd the pointwise Helly selection theorem Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics, University of Toronto Jnury, 015 1 BV [, b] Let < b. For f

### Fundamental Theorem of Calculus for Lebesgue Integration

Fundmentl Theorem of Clculus for Lebesgue Integrtion J. J. Kolih The existing proofs of the Fundmentl theorem of clculus for Lebesgue integrtion typiclly rely either on the Vitli Crthéodory theorem on

### 1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.

Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the

### Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED

Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type

### x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

### Euler-Maclaurin Summation Formula 1

Jnury 9, Euler-Mclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z,

### IMPORTANT THEOREMS CHEAT SHEET

IMPORTANT THEOREMS CHEAT SHEET BY DOUGLAS DANE Howdy, I m Bronson s dog Dougls. Bronson is still complining bout the textbook so I thought if I kept list of the importnt results for you, he might stop.

### Lecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)

Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

### Now, given the derivative, can we find the function back? Can we antidifferenitate it?

Fundmentl Theorem of Clculus. Prt I Connection between integrtion nd differentition. Tody we will discuss reltionship between two mjor concepts of Clculus: integrtion nd differentition. We will show tht

### Properties of the Riemann Integral

Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2

### Main topics for the First Midterm

Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

### 1. On some properties of definite integrals. We prove

This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

### 63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

### UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

### We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.

Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn

### Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

### Homework 11. Andrew Ma November 30, sin x (1+x) (1+x)

Homewor Andrew M November 3, 4 Problem 9 Clim: Pf: + + d = d = sin b +b + sin (+) d sin (+) d using integrtion by prts. By pplying + d = lim b sin b +b + sin (+) d. Since limits to both sides, lim b sin

### 7 Improper Integrals, Exp, Log, Arcsin, and the Integral Test for Series

7 Improper Integrls, Exp, Log, Arcsin, nd the Integrl Test for Series We hve now ttined good level of understnding of integrtion of nice functions f over closed intervls [, b]. In prctice one often wnts

### Appendix to Notes 8 (a)

Appendix to Notes 8 () 13 Comprison of the Riemnn nd Lebesgue integrls. Recll Let f : [, b] R be bounded. Let D be prtition of [, b] such tht Let D = { = x 0 < x 1

### Math 61CM - Solutions to homework 9

Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ

### Analysis III. Ben Green. Mathematical Institute, Oxford address:

Anlysis III Ben Green Mthemticl Institute, Oxford E-mil ddress: ben.green@mths.ox.c.uk 2000 Mthemtics Subject Clssifiction. Primry Contents Prefce 1 Chpter 1. Step functions nd the Riemnn integrl 3 1.1.

### 1 The Riemann Integral

The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

### Entrance Exam, Real Analysis September 1, 2009 Solve exactly 6 out of the 8 problems. Compute the following and justify your computation: lim

1. Let n be positive integers. ntrnce xm, Rel Anlysis September 1, 29 Solve exctly 6 out of the 8 problems. Sketch the grph of the function f(x): f(x) = lim e x2n. Compute the following nd justify your

### f p dm = exp Use the Dominated Convergence Theorem to complete the exercise. ( d φ(tx))f(x) dx. Ψ (t) =

M38C Prctice for the finl Let f L ([, ]) Prove tht ( /p f dm) p = exp p log f dm where, by definition, exp( ) = To simplify the problem, you my ssume log f L ([, ]) Hint: rewrite the left hnd side in form

### Math 360: A primitive integral and elementary functions

Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

### MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

### Math 116 Calculus II

Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................

### Calculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties

Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwth-chen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:

### arxiv: v1 [math.ca] 7 Mar 2012

rxiv:1203.1462v1 [mth.ca] 7 Mr 2012 A simple proof of the Fundmentl Theorem of Clculus for the Lebesgue integrl Mrch, 2012 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde

### MAA 4212 Improper Integrals

Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which

### 0.1 Properties of regulated functions and their Integrals.

MA244 Anlysis III Solutions. Sheet 2. NB. THESE ARE SKELETON SOLUTIONS, USE WISELY!. Properties of regulted functions nd their Integrls.. (Q.) Pick ny ɛ >. As f, g re regulted, there exist φ, ψ S[, b]:

### MATH1013 Tutorial 12. Indefinite Integrals

MATH Tutoril Indefinite Integrls The indefinite integrl f() d is to look for fmily of functions F () + C, where C is n rbitrry constnt, with the sme derivtive f(). Tble of Indefinite Integrls cf() d c

### Math 120 Answers for Homework 13

Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing

### Chapters 4 & 5 Integrals & Applications

Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions

### Big idea in Calculus: approximation

Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

### The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

### FUNDAMENTALS OF REAL ANALYSIS by. III.1. Measurable functions. f 1 (

FUNDAMNTALS OF RAL ANALYSIS by Doğn Çömez III. MASURABL FUNCTIONS AND LBSGU INTGRAL III.. Mesurble functions Hving the Lebesgue mesure define, in this chpter, we will identify the collection of functions

### LECTURE. INTEGRATION AND ANTIDERIVATIVE.

ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development

### For a continuous function f : [a; b]! R we wish to define the Riemann integral

Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This

### Some estimates on the Hermite-Hadamard inequality through quasi-convex functions

Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper

### Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

### Math Calculus with Analytic Geometry II

orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

### The Fundamental Theorem of Calculus, Particle Motion, and Average Value

The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

### Fourier series. Preliminary material on inner products. Suppose V is vector space over C and (, )

Fourier series. Preliminry mteril on inner products. Suppose V is vector spce over C nd (, ) is Hermitin inner product on V. This mens, by definition, tht (, ) : V V C nd tht the following four conditions

### Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

### Continuous Random Variables

STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

### Functions of bounded variation

Division for Mthemtics Mrtin Lind Functions of bounded vrition Mthemtics C-level thesis Dte: 2006-01-30 Supervisor: Viktor Kold Exminer: Thoms Mrtinsson Krlstds universitet 651 88 Krlstd Tfn 054-700 10

### Math Solutions to homework 1

Mth 75 - Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht

### Riemann is the Mann! (But Lebesgue may besgue to differ.)

Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

### The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

### Final Exam - Review MATH Spring 2017

Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.

### different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).

Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different