1549901369-Elements_of_Real_Analysis__Denlinger_

7.4 Basic Existence and Additivity Theorems 391 Definition 7.4.11 A bounded function f: [a, b] --+JR is said to be piecewise con ...

392 Chapter 7 a The Riemann Integral y y =f(x) a b x Figure 7.7 Proof. Exercise 14. • *REGULATED FUNCTIONS While each of the con ...

7.4 Basic Existence and Additivity Theorems 393 Examples 7.4.17 (a) The function f(x) = x - 1 1 x ' is regulated on [O, 5], and ...

394 Chapter 7 • The Riemann Integral If we define the step function CJ^1 on [a, b] by { CT(x) on [a, c - 8], } CJ'(x) = L' on (c ...

7.4 Basic Existence and Additivity Theorems 395 Corollary 7.4.19 If f is a regulated function on [a, b], then f is integrable on ...

396 Chapter 7 • The Riemann Integral { lx 2 ll .f =J 1 } Draw the graph of the function f(x) = x - 1 1 x on the in- 0 if x = ...

7.5 Algebraic Properties of the Integral 397 7.5 Algebraic Properties of the Integral You may have noticed a pattern in earlier ...

398 Chapter 7 • The Riemann Integral (c) If f is integrable on [a, b] and \:Ix E [a, b], lf(x)I ::::; M, then 11: fl < M(b-a) ...

7 .5 Algebraic Properties of the Integral 399 By Riemann's condition for integrability (7.2.14) there is a partition 'P = { Xo, ...

400 Chapter 7 • The Riemann Integral (c) Finally, putting together the results of (16) and (19), we have n I; (Mi(g of) - mi(g o ...

7.5 Algebraic Properties of the Integral 401 Corollary 7.5.7 (Algebra of the Integral, VI-Products and Max/Min) If f and g are i ...

402 Chapter 7 • The Riemann Integral Suppose f and g are continuous on [a, b] and l: f = l: g. Prove that 3 c E [a, b] 3 f(c) = ...

7.6 The Fundamental Theorem of Calculus 403 INTEGRATING DERIVATIVES Definition 7.6.1 A function F is an antiderivative of a func ...

404 Chapter 7 • The Riemann Integral To say this another way, Theorem 7.6.2 says that under certain conditions, J: f' = f(b) - f ...

7.6 The Fundamental Theorem of Calculus 405 But Ica f = -I: f and I: f = -I: f. Thus, Equation (21) becomes I: f =I: f - I: f , ...

406 Chapter 7 • The Riemann Integral E: Let E: > 0. Choose f> = M. Then f> > 0, and Vx, y E J , IY - xi < f> = ...

7.6 The Fundamental Theorem of Calculus 407 Proof. Suppose f is integrable on a compact interval I , and a E J. Define the funct ...

408 Chapter 7 • The Riemann Integral F(x) - F(xo). , Therefore, lim = f(xo). That is, F (xo) = f(xo). • x-+xo X - X o Example 7. ...

7.6 The Fundamental Theorem of Calculus 409 A few words about this notation are in order. The justification for using the "integ ...

410 Chapter 7 • The Riemann Integral on [a, b], u[a, b] is a closed interval I containing c and d (see Corollary 5.3.12). Since ...