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Macroeconomics Exam Review 76

Macroeconomics Exam Review 76 - Solutions for Foundations...

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2.78 Choose any 𝑥 0 𝑋 and 𝜖 > 0. Since 𝑓 is continuous, there exists 𝛿 1 such that 𝜌 ( 𝑥, 𝑥 0 ) < 𝛿 1 = ⇒ ∣ 𝑓 ( 𝑥 ) 𝑓 ( 𝑥 0 ) < 𝜖/ 2 Similarly, there exists 𝛿 2 such that 𝜌 ( 𝑥, 𝑥 0 ) < 𝛿 2 = ⇒ ∣ 𝑔 ( 𝑥 ) 𝑔 ( 𝑥 0 ) < 𝜖/ 2 Let 𝛿 = min { 𝛿 1 , 𝛿 2 } . Then, provided 𝜌 ( 𝑥, 𝑥 0 ) < 𝛿 ( 𝑓 + 𝑔 )( 𝑥 ) ( 𝑓 + 𝑔 )( 𝑥 0 ) = 𝑓 ( 𝑥 ) + 𝑔 ( 𝑥 ) 𝑓 ( 𝑥 0 ) 𝑔 ( 𝑥 0 ) ≤ ∣ 𝑓 ( 𝑥 ) 𝑓 ( 𝑥 0 ) + 𝑔 ( 𝑥 ) 𝑔 ( 𝑥 0 ) < 𝜖 This establishes 𝑓 + 𝑔 is continuous at 𝑥 0 . Since 𝑥 0 was arbitrary, 𝑓 + 𝑔 is continuous for every 𝑥 0 𝑋 . The continuity of 𝛼𝑓 is shown similarly. 2.79 Choose any 𝑥 0 𝑋 . Given 0 < 𝜂 1, there exists 𝛿 > 0 such that 𝑓 ( 𝑥 ) 𝑓 ( 𝑥 0 ) < 𝜂 and 𝑔 ( 𝑥 ) 𝑔 ( 𝑥 0 ) < 𝜂 whenever 𝜌 ( 𝑥, 𝑥 0 ) < 𝛿 . Consequently, while 𝜌 ( 𝑥, 𝑥 0 ) < 𝛿 𝑓 ( 𝑥 ) ∣ ≤ ∣ 𝑓 ( 𝑥 ) 𝑓 ( 𝑥 0 ) + 𝑓 ( 𝑥 0 ) < 𝜂 + 𝑓 ( 𝑥 0 ) 1 + 𝑓 ( 𝑥 0
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