Lecture20101021_1

Lecture20101021_1 - Numerical Analysis II Dr Abigail Wacher...

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Unformatted text preview: Numerical Analysis II Dr Abigail Wacher 1 Aside Continuity Recall that f: ℝ-> ℝ is continuous if ∀ ε > 0 ∃ δ > 0 s.t. |x-y|< δ => |f(x)-f(y)|< ε But some functions are "more continuous" than others, so we need a more refined definition. Definitions : Let I ⊂ = d , then f:I->I is - Lipschitz (cts) if ∃ L ϵ ℝ + s.t. ∀ x,y ϵ I, |f(x)-f(y)| ≤ L|x-y|- Hölder (cts) if ∃ L ϵ ℝ + s.t. ∀ x,y ϵ I, |f(x)-f(y)| ≤ L x K y a with α ϵ (0,1). Examples I= ℝ f(x) cts Hölder Lipschitz Differentiable 2 yes no no yes sin x yes yes yes (L=1) yes |sinx| yes yes yes (L=1) no sinx yes yes ( α =1/2) no no 1 3 yes yes ( α =1/2) no no I= K 10 100 , 10 100 f(x) cts Hölder Lipschitz Differentiable 2 yes yes yes (L=4.10E100) yes sin x yes yes yes yes |sinx| yes yes yes no yes yes ( α =1/2) no no 1 3 yes yes ( α =1/2) no no Notation: f cts in I: f ϵ C I f Lipschitz: f ϵ 0, 1 f Hölder: f ϵ 0, a f differentiable: f ϵ 1 Recall that f is differentiable => f is cts OR 1 3 Fact:...
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This note was uploaded on 02/09/2011 for the course MATH 2051 taught by Professor Dr.a.wacher during the Fall '11 term at Durham.

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Lecture20101021_1 - Numerical Analysis II Dr Abigail Wacher...

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