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Unformatted text preview: 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, Newtons 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 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 < 1 if 0 x < 1 . The Dirichlet function : R R given by ( x ) = ( 1 if x Q if x 6 Q . Definition 2.1 Let I be an open interval, f : I R be a function, and x 3 I . We say that f is continuous at x if for every > there exists > such that if x I , | x- x | < then | f ( x )- f ( x ) | < . If a function f is not continuous at x we say that f is discontinuous at x . Proposition 2.2 Let I be an open interval, f : I R be a function, and x 3 I . The following are equivalent: (i) f is continuous at x ; (ii) for every sequence ( x n ) n =1 I with lim n x n = x we have that lim n f ( x n ) = f ( x ) . Examples (i) The constant function is continuous at every point. (ii) The identity function is continuous at every point....
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- Spring '08