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Unformatted text preview:  t i − t i − 1  < ε for each i will work. (c) Prove by contradiction that ε > 0: if not, there exist pairs t ′ i n = t ′′ i with  t ′ i − t ′′ i  < 1 i −→ p ( t ′ i ) = −→ p ( t ′′ i ). By going to a subsequence, assume t ′ i → t . Show that also t ′′ i → t . But then −→ p is not locally onetoone at t . (d) Now, let I be any interval (open, closed, or halfopen). Show that there is a bisequence of points t i , i ∈ Z such that t i − 1 < t i and −→ p is onetoone on [ t i − 1 ,t i ] ( Hint: By the preceding, we can partition any closed bounded subinterval of I into Fnitely many arcs. Use this to show that all of I can be partitioned into a possibly inFnite collection of abutting arcs.) 12. Show that the ±olium of Descartes (±igure 2.25 ) is given by the polar equation r = 3sin θ cos θ sin 3 θ + cos 3 θ ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus, Derivative

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