1549901369-Elements_of_Real_Analysis__Denlinger_

5.7 *sets of Points of Discontinuity (Project) 291 Proof. The following proof is a little sketchy. You are asked to fill in the ...

292 Chapter 5 • Continuous Functions OSCILLATION OF A FUNCTION Before learning more about the sets of discontinuity of functions ...

5.7 *sets of Points of Discontinuity (Project) 293 Definition 5. 7.12 A set is called an F,, set if it is the union of countably ...

294 Chapter 5 • Continuous Functions Comments: A countable set is of the first category, since every single point set { x} is no ...

5.7 *sets of Points of Discontinuity (Project) 295 Lemma 5.7.18 (a) A closed set either contains a proper interval or is nowhere ...

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Chapter 6 Differentiable Functions Sections 6.1-6.3 develop the standard introductory material on derivatives and differentiabil ...

298 Chapter 6 • Differentiable Functions Proof. Let f(x) =ax + b. For arbitrary Xo E IR, lim _f-'---(x--'--)_-_f-'(_xo-'-) = lim ...

6.1 The Derivative and Differentiability 299 Case 2 (xo < 0): Then 1 . f(x) - f(xo) lm 1 . im---lxl - lxol x-+xo x - Xo x-+xo ...

300 Chapter 6 • Differentiable Functions ... f(x) - f(xo) (See Exercise 4.1.6.) Because of this, if we let g(x) = , then we x -x ...

6.1 The Derivative and Differentiability 301 Theorem 6.1.8 says that differentiability is "stronger" than conti- nuity; that is, ...

302 Chapter 6 • Differentiable Functions Therefore, f is differentiable at 0, and f'(O) = 0. D Another caution: Although differe ...

6.1 The Derivative and Differentiability 303 (b) If 3 o > 0 :1 f is differentiable on (xo, xo+o) and continuous from the righ ...

304 Chapter 6 • Differentiable Functions Find the values of the constants a and b for which the function j(x) = { x 2 ~f x:::; ...

6.2 Rules for Differentiation 305 16. (a) Prove that if f is differentiable at x 0 , then lim f(xo + h) ~f(xo - h) h->O 2 exi ...

306 Chapter 6 • Differentiable Functions NOTATION FOR DERIVATIVES It is somewhat cumbersome to continue using the subscript in x ...

6.2 Rules for Differentiation 307 Proof. Suppose f and g are differentiable at xo and c E R Then ( ) 1 . cf(x) -cf(xo) 1 . [f(x) ...

308 Chapter 6 • Differentiable Tunctions Now, f is differentiable at x 0. Hence, by Theorem 6.1.8, f is continuous at x 0. That ...

6.2 Rules for Differentiation 309 Theorem 6.2.3 (The Chain Rule) Suppose f is differentiable at an inte- rior point Xo of its do ...

310 Chapter 6 11 Differentiable Functions Thus, lim (go J)(x) - (go J)(xo) = lim h (f (x)) lim l (x) - l (xo) x->xo x - x 0 x ...