given the function:

f(x,y)= if

f(x,y) =0 if (x,y)=(0,0)

prove whether it is continuous or discintinuous at (0,0) by using the ε-δ definition for continuity

Printable View

- Dec 1st 2010, 07:37 AMalexandrosproving continuity
given the function:

f(x,y)= if

f(x,y) =0 if (x,y)=(0,0)

prove whether it is continuous or discintinuous at (0,0) by using the ε-δ definition for continuity - Dec 1st 2010, 07:52 AMchisigma
The limit...

...*doesn't exist*so that the function is discontinous in ...

Kind regards

- Dec 1st 2010, 08:19 AMalexandros
- Dec 1st 2010, 01:21 PMchisigma
- Dec 1st 2010, 03:09 PMalexandros
How,then do we otherwise

**prove**that the function is discontinuous at (0,0) - Dec 1st 2010, 03:35 PMnerdo
- Dec 1st 2010, 03:37 PMPlato
- Dec 1st 2010, 03:40 PMnerdo
- Dec 1st 2010, 03:42 PMPlato
- Dec 1st 2010, 03:52 PMnerdo
I think personally if one used the theorem below, the question is trivial:

Attachment 19924 - Dec 1st 2010, 03:55 PMPlato
- Dec 1st 2010, 03:59 PMnerdo
Oh ok i see what you mean, i guess you are right.

- Dec 1st 2010, 04:32 PMalexandros
- Dec 1st 2010, 05:12 PMalexandros
To show that :

**does not exist**?

We must show that: for all real Nos m there exists an ε>0 such for all δ>0 there exists a pair (x,y) such that : |x|<δ and |y|<δ and

However to show discontinuity at (0,0) we must show that:

There exists ε>0 such that for all δ>0 there exists (x,y) such that:|x|<δ and |y|<δ and

**Two comletely different things** - Dec 2nd 2010, 12:29 PMSammySproving continuity

I realize that you need an ε-δ proof, but first let's find a sequence that shows that is discontinuous at (0,0). This may help with your ε-δ proof.

The problem with is that it is undefined along the line , except at (0,0). Let's make a sequence which approaches (0,0) along a path that approaches . One such path is . Define , then , so .

Since , we have that is not continuous at (0,0).

**Now for your ε-δ proof:**

Rather than showing that :**does not exist**, just show that it's not 0. Modify your statement to have m=0.

Given any , let (may have to tweak this a little). Let .

Then

Fill in the rest of the details.