# hw8 - Math 341 Homework # 8 P104. 13, 14, 17. P105. 19, 20,...

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Math 341 Homework # 8 P104. 13, 14, 17. P105. 19, 20, 21, 22. 13. Let f : D R be continuous at x 0 D . Prove that there is M > 0 and a neighborhood Q of x 0 such that | f ( x ) | ≤ M for all x Q D . Proof: Since f is continuous at x 0 , ± > 0, δ > 0, for x D with | x - x 0 | < δ , we have | f ( x ) - f ( x 0 ) | < ±. Let ± = 1, M = | f ( x 0 ) | + 1 and Q = ( x 0 - δ,x 0 + δ ). Then for x Q D , | f ( x ) | = | f ( x ) - f ( x 0 ) + f ( x 0 ) | ≤ | f ( x ) - f ( x 0 ) | + | f ( x 0 ) | < 1 + | f ( x 0 ) | = M. 14. If f : D R is continuous at x 0 D , prove that | f | : D R such that | f | ( x ) = | f ( x ) | is continuous at x 0 . Proof: Since f is continuous at x 0 , ± > 0, δ > 0, for x D with | x - x 0 | < δ , we have | f ( x ) - f ( x 0 ) | < ±. Hence || f | ( x ) - | f | ( x 0 ) | ≤ | f ( x ) - f ( x 0 ) | < ±. Thus, | f | is continuous at x 0 . 17. Suppose f : D R with f ( x ) 0 for all x D . Show that, if f is continuous at x 0 , then f is continuous at x 0 . Proof: Since

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## This note was uploaded on 08/13/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.

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hw8 - Math 341 Homework # 8 P104. 13, 14, 17. P105. 19, 20,...

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