Econ617set1re - TA Yankun Wang Econ 617 Problem Set 1 Solution Key 1 The whole point of this problem is to understand the defition of continuity

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TA: Yankun Wang Econ 617 Problem Set 1 Solution Key 1. The whole point of this problem is to understand the de f tion of continuity and to get used to the ε δ argument. (a): For f : R + R + , we say that f is continuous if: for every x R + and for any ε> 0 ,w ecan f nd a δ> 0 , and δ depending on ε, such that for all x satisfying | x x | and x R + ,wehave | f ( x ) f ( x ) | <ε. (i) f ( x )= x : pick an arbitrary x R + , and for any ε> 0 , let δ = ε : the rest should be obvious. (ii) f ( x )=1+ x : again, pick an arbitrary x R + , and let δ = ε : for all x satisfying | x x | and x R + , | f ( x ) f ( x ) | = | x x | = ε. (iii) f ( x )= 1 1+ x : pick an arbitrary x R + , and let ε> 0 .F o ra n y x R + , | f ( x ) f ( x ) | = | 1 1+ x 1 1+ x | = | x x (1 + x )(1 + x ) | = | x x | (1 + x )(1 + x ) . Since 1+ x 1 , 1+ x 1 , we have 0 < 1 (1+ x )(1+ x ) 1 . Now just let δ = ε, for x satisfying | x x | and x R + , | f ( x ) f ( x ) | = ε. To draw of the graph of this function, notice that we can obtain it by shift the graph of f (

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This note was uploaded on 08/29/2009 for the course ECON 617 taught by Professor Staff during the Fall '08 term at Cornell University (Engineering School).

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Econ617set1re - TA Yankun Wang Econ 617 Problem Set 1 Solution Key 1 The whole point of this problem is to understand the defition of continuity

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