Limit and Continuity (multivariable calculus) [help]

**catcat103** · Jun 13th 2008, 07:10 AM

Question : For f(x,y)=xy, show that f(x,y) is continuous everywhere.
i.e. for every point (a,b) in the plane and for every epsilon>0, find delta>0 depending on a,b and epsilon such that whenever
0<|(x,y)-(a,b)|<delta, |f(x,y)-f(a,b)|<epsilon

How to find an epsilon in terms of delta? I am troubled in expressing epsilon as a function of delta.

Many thanks.

**kalagota** · Jun 13th 2008, 07:14 PM

Originally Posted by catcat103

Question : For f(x,y)=xy, show that f(x,y) is continuous everywhere.
i.e. for every point (a,b) in the plane and for every epsilon>0, find delta>0 depending on a,b and epsilon such that whenever
0<|(x,y)-(a,b)|<delta, |f(x,y)-f(a,b)|<epsilon

How to find an epsilon in terms of delta? I am troubled in expressing epsilon as a function of delta.

Many thanks.

epsilon is an arbitrary number.. you should find delta in terms of epsilon and not the other way around..

**ThePerfectHacker** · Jun 14th 2008, 06:41 PM

Originally Posted by catcat103

Question : For f(x,y)=xy, show that f(x,y) is continuous everywhere.

We shall prove $f$ is continous at $(a,b)$ . Let $\sqrt{(x-a)^2+(y-b)^2} < \delta$ . Then $|x-a| = \sqrt{(x-a)^2}\leq \sqrt{(x-a)^2+(y-b)^2} < \delta$ . Similarly $|y-b| < \delta$ . Now use the inequality $||y|-|b|| \leq |y-b| < \delta \implies - \delta < |y| - |b| < \delta$ . Thus, $|y| < \delta + |b|$ . With these facts we can prove continuity. We need to show $|xy-ab|$ can be made arbitrary small. Add and subtact, $|y(x-a) + a(y-b)| \leq |y||x-a| + |a||y-b| < (\delta + |b|)\delta + |a| \delta < \delta (1+|a|+|b|)$ if $0<\delta \leq 1$ .
This tells us that for any $\epsilon > 0$ choose $\delta = \min \left\{ 1 , \tfrac{\epsilon}{1+|a|+|b|} \right\}$ .

**catcat103** · Jun 17th 2008, 04:37 AM

ah...i forgot those useful inequalities. what a marvellous answer! thanks for your reply!

Thread: Limit and Continuity (multivariable calculus) [help]

LinkBack

Thread Tools

Display

Limit and Continuity (multivariable calculus) [help]

Similar Math Help Forum Discussions

Continuity of multivariable function

continuity of multivariable functions

Limit for multivariable calculus

Can you Teach Yourself Calculus 2 (Integral) and Calculus 3 (Multivariable)

calc help (limit and continuity of a multivariable equation)

Search Tags