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LampShade

Member

Registered: 2009-02-22

Posts: 23

Differentiation in Real Analysis

I'm trying to study Real Analysis and I need a bit of help on a problem.  This is number 7 on page 167 of Bartle's and Sherbert's Introduction to Real Analysis, 3rd Edition.

Suppose that

is differentiable at

and that

.  Show that

is differentiable at

if and only if

.

Last edited by LampShade (2009-02-22 18:30:42)

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--  Boozer

LampShade

Member

Registered: 2009-02-22

Posts: 23

Re: Differentiation in Real Analysis

I think I have a solution, but I'm not sure about my reasoning at the end.

Let

.  Then

.

By the chain rule,

.

This implies that

.

If

this limit is undefined since

.

If

, then

is defined and is 0.

I'm just not sure why

removes the discontinuity created by dividing by

.

Last edited by LampShade (2009-02-23 13:56:20)

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--  Boozer

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Differentiation in Real Analysis

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Differentiation in Real Analysis

It is rare that you can use anything from calculus in an analysis class.  Typically you are trying to prove results that reach beyond the basic theory.  With that in mind, it is never a bad idea to write down precisely what these things means.

Try relating the two.  It will be more concrete to use epsilon-delta definitions.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

LampShade

Member

Registered: 2009-02-22

Posts: 23

Re: Differentiation in Real Analysis

Ok!  I posted this question on Yahoo! answers, and I got a reply there as well.  I took Rick's advice and combined it with the reply from Yahoo to get the following.  Please let me know if there are any mistakes.  Oh, and thanks for the warning about the chain rule!!!

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--  Boozer