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material2 - Material for PMA104 Ivan Todorov March 5 2008...

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Material for PMA104 Ivan Todorov March 5, 2008 The file contains the material covered in PMA104. The statements are proved in class unless otherwise stated. 1 Preliminaries Sets, operations with sets, properties of the operations with sets. The sets N , Z , Q , R , C . Notation for intervals: [ a, b ] for closed intervals, ( a, b ) for open intervals etc. Mappings between sets. Injective, surjective and bijective mappings. Sequences of real numbers. Convergence of sequences. Arithmetics of limits: lim n →∞ ( a n + b n ) = lim n →∞ a n + lim n →∞ b n etc. The Sandwich Rule. Basic inequalities and identities: the triangle inequality, Newton’s Bino- mial Formula, a n - b n = ( a - b )( a n - 1 + a n - 2 b + · · · + b n - 1 ). 2 Analysis 2.1 Subsequences Definition of a subsequence. Notation: ( a n k ) k =1 . Every subsequence of a convergent sequence converges to the same limit. Examples: ( 1 2 n ), ( 1 n 2 ). Definition of a function f : I R . The domain is always an interval. The definition of the graph of a function. Review of basic functions and how 1
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their graphs look like: the constant and the identity functions, x n , n x , a x ( a > 1), log x , sin x , cos x , tan x , cot x . The step function f : ( - 1 , 1) R given by f ( x ) = ( - 1 if - 1 < x < 0 1 if 0 x < 1 . The Dirichlet function χ : R R given by χ ( x ) = ( 1 if x Q 0 if x 6∈ Q . Definition 2.1 Let I be an open interval, f : I R be a function, and x 0 3 I . We say that f is continuous at x 0 if for every ² > 0 there exists δ > 0 such that if x I , | x - x 0 | < δ then | f ( x ) - f ( x 0 ) | < ² . If a function f is not continuous at x 0 we say that f is discontinuous at x 0 . Proposition 2.2 Let I be an open interval, f : I R be a function, and x 0 3 I . The following are equivalent: (i) f is continuous at x 0 ; (ii) for every sequence ( x n ) n =1 I with lim n →∞ x n = x 0 we have that lim n →∞ f ( x n ) = f ( x 0 ) . Examples (i) The constant function is continuous at every point. (ii) The identity function is continuous at every point. (iii) The function f : R R given by f ( x ) = λx , x R , where λ is a fixed real number, is continuous at every point.
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