18.100pset8

18.100pset8 - 18.100B/C: Fall 2008 Solutions to Homework 8...

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18.100B/C: Fall 2008 Solutions to Homework 8 1. (a) Dividing the given equation through by | x - y | gives 0 | f ( x ) - f ( y ) | | x - y | C · | x - y | α - 1 . As y x , the right-hand term in this inequality approaches 0, so the middle term must approach 0 as well and we have lim y x | f ( x ) - f ( y ) | | x - y | = 0 . Since f ( x ) - f ( y ) x - y = ± | f ( x ) - f ( y ) | | x - y | for every x 6 = y , this implies that f 0 ( x ) exists for each x and that f 0 ( x ) = 0 for all x . Then by Rudin 5.11(a), f ( x ) must be a constant function, as desired. (b) No. For example, for any α (0 , 1], consider the function f α ( x ) = | x | α . This function satisﬁes | f α ( x ) - f α ( y ) | ≤ | x - y | α for all x,y R but is not diﬀerentiable at 0 for any α 1. 2.(a) By considering the closed interval [ x,y ], it follows immediately from the Mean Value Theorem that if | f 0 ( x ) | ≤ M for all x then | f ( x ) - f ( y ) | ≤ M ·| x - y | for all x,y (i.e., functions with bounded derivative are Lipschitz). We will show that Lipschitz functions take Cauchy sequences to Cauchy sequences: suppose ( x n ) n N is a Cauchy sequence. Then for any ε > 0 we can ﬁnd N N such that m,n N implies | x m - x n | < ε M . Then it follows immediately that for all m,n N we have | f ( x m ) - f ( x n ) | < M · ε M = ε , so ( f ( x n )) n N is Cauchy. In particular, ( f ( 1 n ) ) n N is Cauchy and so converges in R , as needed. (b) As mentioned in part (a), we have by the Mean Value Theorem that if | f 0 ( x ) | ≤ α for all x then f is Lipschitz with constant α . If α < 1 then by problem set 6, problem 7 f has a unique ﬁxed point, as desired. (c) No. Consider for example the function f ( x ) = x 2 + 1. We have f ( x ) > x for all x R , so f has no ﬁxed points, but f 0 ( x ) ( - 1 , 1) for all x R . (This function is half of a hyperbola with asymptotes y = x and y = - x .) 3. (Rudin page 114 problem 2) Suppose

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This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.

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18.100pset8 - 18.100B/C: Fall 2008 Solutions to Homework 8...

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