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Unformatted text preview: Section 5.1 #1) By the definition of continuity, f is continuous at c if and only if lim x c f = f ( c ). By the Sequential Criterion in Chapter 4, this is equivalent to the statement that if ( x n ) is a sequence converging to c such that x n 6 = c , then ( f ( x n )) converges to f ( c ). Clearly, this condition is satisfied if the Sequential condition in 5.1.3 (which allows for ANY sequence converging to c ) is satisfied. Conversely, let ( x n ) be any sequence converging to c . Either it is ultimately constant and equal to c , and there is nothing to prove since f ( x n ) will be ultimately constantly f ( c ), or the terms x n k not equal to c form an infinite subsequece converging to c . Thus, if f is continuous at c , f ( x n k ) converges to f ( c ), and hence f ( x n ) converges to f ( c ), since every term not in f ( x n k ) is equal to f ( c ). #3) Let &gt; 0. If x [ a, b ), then we can find &gt; 0 such that x + &lt; b , and if x [ a, b ], | x- x | &lt; , then | f ( x )- f ( x ) | &lt; . (Note that if x [ a, c ], | x- x | &lt; , then automatically, x [ a, b ).) Since h...
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This note was uploaded on 09/30/2009 for the course MATH 444 taught by Professor Junge during the Spring '08 term at University of Illinois at Urbana–Champaign.
- Spring '08