1549901369-Elements_of_Real_Analysis__Denlinger_

5.5 *Monotonicity, Continuity, and Inverses 271 REMARKS: (1) 'Pc(x) is the binary expansion of a unique real number in [O, l]. ( ...

272 Chapter 5 • Continuous Functions where xi, Yi E {O, 2}. Let n denote the first (smallest) natural number such that Xn -=f. Y ...

5.5 *Monotonicity, Continuity, and Inverses 2 7 3 y 1 7 8 3 4 5 8 I 2 3 8 I 4 l 8 I 2 I 2 7 8 x (^9 9 3 3 9 9) Fig ure 5.1 2 The ...

27 4 Chapter 5 • Continuous Functions EXERCISE SET 5.5 Prove Lemma 5.5.1. Prove that a function f is monotone (or strictly) inc ...

5.5 *Monotonicity, Continuity, and Inverses 275 (b) 1 is { positive and strictly decreasing on ( -oo, 0) if n is even; }. negati ...

276 Chapter 5 • Continuous Tunctions (c) Let r E Q. Prove that if r > 0, the function f(x) = xr is positive, continuous, and ...

5.6 *Exponentials, Powers, and Logarithms 277 show that loga x = ln x / ln a. In a later chapter of elementary calculus, power s ...

278 Chapter 5 • Continuous Functions exponent) to derive an important property of the function f(x) =ax (constant base). Thus, t ...

5.6 *Exponentials, Powers, and Logarithms 279 any rational number greater than x, then \:/n E N, rn :::; x < r, so arn < a ...

280 Chapter 5 • Continuous Tunctions Reversing the roles of { r n} and {Sn} will allow us to prove L :::; M. There- fore, L = M. ...

5.6 *Exponentials, Powers, and Logarithms 281 (3) Let x E JR. In Definition 5.6.5, the sequence { arn} is a monotone in- creasin ...

282 Chapter 5 • Continuous Functions Lemma 5.6.9 Let a ;::: 1 and x E R If {tn} is any monotone decreasing sequence of rational ...

5.6 *Exponentials, Powers, and Logarithms 283 Theorem 5.6.11 (Exponential Functions, I) For a > 1, the exponential function f ...

284 Chapter 5 • Continuous Functions Since limits preserve inequalities, xt :=::; qi :=::; q~ :=::; yt. By Exercise 5.5.15, qi & ...

5.6 *Exponentials, Powers, and Logarithms 285 Hence sup{ xt : x < xo} > Xb - c. By the forcing principle, this implies sup ...

286 Chapter 5 11 Continuous Functions Theorem 5.6.15 (Negative Power Functions) Let t < 0. The power function f ( x) = xt, de ...

5.6 *Exponentials, Powers, and Logarithms 287 function f ( x) = ( 1 + ~ r. To make this transition we will use the "greatest int ...

288 Chapter 5 11 Continuous Functions 5.6.14 and 5.6.15; moreover, by Lemma 5.6.19, lim g(t) = e. By Theorem 5.1.14, t-+O lim(l ...

5.6 *Exponentials, Powers, and Logarithms 289 Of course, there are many possible bases to use in logarithm functions; in fact, a ...

290 Chapter 5 • Continuous Functions 7 *sets of Points of Discontinuity (Project) This section is icing on the cake. It answer ...