bmhicks_homework_3

bmhicks_homework_3 - Brandy Hicks September 12, 2007 MTHSC...

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September 12, 2007 MTHSC 974.001 - Homework # 3 1. Prove that if f con V ( R ), then its domain must be an interval. Hint: Suppose t d om ( f ) and it is isolated. Then an r > 0 3 ( t - r,t ) ( t,t + r ) / d om ( f ) and s > t + r, s d om ( f ). You should be able to show the convex inequality for f doesn’t work for appropriate points. Proof: Let f con V ( R ) = { f : R R ∪ { + ∞}} . Let t d om ( f ) and let t be isolated. Then, r > 0 3 ( t - r,t ) ( t,t + r ) and r d om ( f ) where r R Now, noting ± t + t + r 2 ² ( t - r,t ) ( t,t + r ) and ± t + t + r 2 ² / d om ( f ) , then f ± t + t + r 2 ² = { + ∞} Then s d om ( f ) 3 s > ± t + t + r 2 ² > t Now since f con V ( R ) and s > ± t + t + r 2 ² > t Then f ± t + t + r 2 ² < αf ( t ) + (1 - α ) f ( s ) for some α (0 , 1) But this clearly can’t be true since f ± t + t + r 2 ² = { + ∞} Then clearly d om ( f ) can not contain an isolated element. d
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bmhicks_homework_3 - Brandy Hicks September 12, 2007 MTHSC...

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