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# Real Analysis MAT 125A

UCD

GPA 3.88

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This 3 page Study Guide was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Study Guide belongs to MAT 125A at University of California - Davis taught by Staff in Fall. Since its upload, it has received 35 views. For similar materials see /class/187456/mat-125a-university-of-california-davis in Mathematics (M) at University of California - Davis.

## Reviews for Real Analysis

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Date Created: 09/08/15

THINGS YOU MUST KNOW This list highlights everything you have learned over the quarter In some cases I7ve written you 77must know the proof I do not mean you have memorize it but you should be able to 77sketch77 it It is important to know the statements of the theorems and is also important to know how to apply them to solve problems Section 17 a 676 and sequential de nition of continuity You must know how to prove the equivalence between these two de nitions b Prove that functions are continuous using the de nition c Provide examples of discontinuous functions d Operations of continuous functions7 ief gfgfg7 9 You must know these proofs Also f o 9 Section 18 Properties of continuous functions a Theorem 181 maximum minimum b Intermediate Value Theorem Section 19 Uniform Continuity a De nition of uniform continuity b Theorem If f continuous on 11 then f is uniformly con tinuous look at the proof7 the argument is standard and used in several other proofs c Theorem f uniformly continuous on S and 5 is a Cauchy sequence in S then fsn is a Cauchy sequence another standard proof you must know You also need to know how to use the reciprocal of this theorem d Theorem 195 e Theorem 196 Section 20 Limits of functions a Sequential and E 7 6 de nitions of limits of functions Two sided limits7 one sided limits b Recall In the expresion limwns z L7 5 is a symbol a7 cf7 1 ioo7 a E R And L can be nite or ioo Section 23 Power series a De ntion of power series b Convergence of power series7 radius of convergence7 interval of convergence Theorem 231 Section 24 Uniform convergence A 1 Section 25 Section 26 Section 28 Section 29 THINGS YOU MUST KNOW a De nition of pointwise convergence Find pointwise limit for a sequence of functions b De nition of uniform convergence Determine if a sequence of functions converges uniformly to its pointwise limit c Theorem 243 The uniform limit of continuous functions is continuous Apply its reciprocal7 as well d Remark 244 fn a sequence in S converges uniformly to f if and only if limnaoosuplfz A z E 5 0 More on Uniform convergence a Theorem 252 If fn A f uniformly on 1 b then limyH00 fba fnzdx ffxd Apply its reciprocal b De nition fn is uniformly Cauchy on S c Theorem 254 d Application to Series of functions Theorem 2557 256 Proofs are easy7 you must know them e Weierstrass M test Apply it to determine if a series of functions converges uniformly Differentiation and Integration of Power Series The main point here is to prove that the derivative of a power series is the series ofthe derivatives and the integral ofthe power series is the series of the integrals a Take a look at Theorems 2617 2627 2637 2647 265 b Abel7s Theorem You must know to apply it Differentiation De nition of derivative Find derivatives of functions using the de nition Theorem If f is differentiable at a point 17 then f is con tinuous at a You must know the proof of this theorem d Example of a function showing the converse of the theorem is not true7 ie f continuous at a is not differentiable at a e Operations with derivatives f g 7 fg 7 Chain rule7 etc The Mean Value Theorem Theorem 291 Rolle7s Theorem good idea to know how to prove it Mean Value Theorem another nice proof to remind d Corollaries 2947 2957 297 AA AAA 0 C7 in VVV lntermediate Value Theorem for derivatives VERY IMPORTANT Theorem 299 Derivative of the in verse function not need to prove it7 but must know how to apply it THINGS YOU MUST KNOW 3 Section 30 L7Hospital7s Rule Make sure you know in which cases you can apply itl7 Check example 4 for a wrong application of L7Hospital7s Rule You do not need to know the proof of the theorem Section 31 Taylor7s Theorem a De nition of Taylor Series for a function f about 0 b De nition of the Remainder c Super lmportant k f Eziofomxk if and only if lim 0 d Version 1 Taylor7s Theorem theorem 313 and Corollary 14 e Version 2 Taylor7s Theorem theorem 315 and corollary 316 f Binomial Series Theorem g You must know how to nd Taylor Series of functions and determine if it equals the function Section 21 Continuity on Metric Spaces a Continuity and Uniformly continuity on a metric space b Proposition 2127 Theorem 2137 Theorem 2147 Corollary

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