JhalyssaWilliams.Analysis.HW7 - Section 5.4 2a >0 >0 |f x)f y)|< whenever| x y|< x y D False depends on on the point x y If > 2b True By definition

JhalyssaWilliams.Analysis.HW7 - Section 5.4 2a >0 >0 |f x)f...

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Section 5.4 2a) ∀ ϵ > 0, δ > 0 | f ( x ) f ( y ) | < ϵ whenever | x y | < δ x, y D .False ;δ depends on ϵ onthe point ( x , y ) . If ϵ > 2b) True. By definition 5.4.1, every continuous function is uniformly continuous f is uniformly continuous on set D. Let (x n ) be a Cauchy sequence in D and given ϵ > 0 , (since f is uniformly contin. On D) , δ > 0 for every point (x,y) D such that |x-y| < δ |f(x)-f(y)|< ϵ . 2c) True, by theorem 5.4.1 3d) f(1) is defined and f(4) is defined the domain can be extended to [1,4] By theorem 5.4.9, f(x) is uniformly continuous and f:[1,4] -> R is continuous and compact. 4a) Let a,b [0,2] and |f(a)-f(b)|=a 3 -b 3 = |a-b||(a 2 +ab+b 2 )|≤ |a-b|(|a 2 |+|ab|+|b 2 |) |f(a)-f(b)|≤ |a-b|(|a 2 |+|ab|+|b 2 |) |f(a)-f(b)|≤ |a-b|(12), where ϵ = 12|(a-b)| Now let δ = ϵ 12 then |f(a)-f(b)|< ϵ whenever |a-b| < δ f is uniformly continuous on [0,2] 5) Let ϵ > 0 and a,b [0, ). Suppose δ = ϵ 2 . Then | a b a + b ¿ | a b | < δ = ϵ 2 | a b ¿ 2 | a b ¿ a + b |= | a b | < ϵ 2 | a b ¿ ϵ f is uniformly continuous on [0, ) 8a)
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  • Fall '08
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  • Calculus, lim, Continuous function, Uniform continuity, 1

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