solutions3

# solutions3 - CM221A ANALYSIS I Solutions to Sheet 3 1 Since...

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CM221A ANALYSIS I Solutions to Sheet 3 1. Since the sums and products of continuous functions are continuous functions, it is suﬃcient to prove the statement for x n , where n is a non-negative integer. Let us ﬁx c R and show that x n is continuous at c . We have c n - x n = ( x - y ) P ( c,x ) where P ( c,x ) = c n - 1 + c n - 2 x + ... + cx n - 2 + x n - 1 . If | c - x | < δ and δ < 1 then | x | < | c | + 1, | P ( c,x ) | < | c | n - 1 + | c | n - 2 | x | + ... + | c || x | n - 2 + | x | n - 1 n ( | c | + 1) n - 1 and, consequently, | c n - x n | < δ n ( | c | + 1) n - 1 . It follows that | c n - x n | < ε whenever | c - x | < δ and the positive number δ satisﬁes the inequalities δ < 1 and δ < εn - 1 ( | c | + 1) 1 - n . Thus, for any ε > 0 we can ﬁnd δ > 0 such that | x - x | < δ ⇒ | c n - x n | < ε , which means that x n is continuous at c . 2. If c (0 , + ) then | x - c | = | x - c | ( x + c ) - 1 . If c > 0 then the right hand side is estimated by c - 1
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