Lecture20101021_1

Lecture20101021_1 - Numerical Analysis II Dr Abigail Wacher...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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:...
View Full Document

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.

Page1 / 2

Lecture20101021_1 - Numerical Analysis II Dr Abigail Wacher...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online