Lecture20101021_1

Lecture20101021_1 - Numerical Analysis II Dr Abigail Wacher...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
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: -> is continuous if > 0 > 0 s.t. |x-y|< => |f(x)-f(y)|< 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 + s.t. x,y I, |f(x)-f(y)| L|x-y|- Hlder (cts) if L + s.t. x,y I, |f(x)-f(y)| L x K y a with (0,1). Examples I= f(x) cts Hlder 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 ( =1/2) no no 1 3 yes yes ( =1/2) no no I= K 10 100 , 10 100 f(x) cts Hlder Lipschitz Differentiable 2 yes yes yes (L=4.10E100) yes sin x yes yes yes yes |sinx| yes yes yes no yes yes ( =1/2) no no 1 3 yes yes ( =1/2) no no Notation: f cts in I: f C I f Lipschitz: f 0, 1 f Hlder: f 0, a f differentiable: f 1 Recall that f is differentiable => f is cts OR 1 3 Fact:...
View Full Document

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 Right Arrow Icon
Ask a homework question - tutors are online