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
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
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.