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solHMK9 - Section 5.1#1 By the denition of continuity f is...

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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 = 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 > 0. If x 0 [ a, b ), then we can find δ > 0 such that x 0 + δ < b , and if x [ a, b ], | x - x 0 | < δ , then | f ( x ) - f ( x 0 ) | < . (Note that if x [ a, c ], | x - x 0 | < δ , then automatically, x [ a, b ).) Since h f on [ a, b ), then for x [ a, c ], if | x - x 0 | < δ , then | h ( x ) - h ( x 0 ) | = | f
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