Show whether f(x,y) is continuous at (0,0)

[Erin0702]

3. Let f(x,y)=( (x²+y²) / (x²+y²) )
a) Find lim f(x,y) as (x,y) approaches (0,0)

lim as (x,y) approaches (0,0) =((x²+y²)/(x²+y²))=1

b) Show whether or not f(x,y) is continuous at (0,0).

I was able to find the answer to the first part of the question...but I'm not exactly sure how to show the second part. Do I need to go through each of the rules and show if it is true or false?

Thanks![/tex][/quote]

 

[pka]

3. Let f(x,y)=( (x²+y²) / (x²+y²) )
b) Show whether or not f(x,y) is continuous at (0,0).
[/tex]

[/quote]
Well of course, f(x,y) is not even define at (0,0)!
To be continuous, must it be defined?

Suppose we have: \(\displaystyle f(x,y) = \left\{ \begin{array}{l}
\frac{{x^2 + y^2 }}{{x^2 + y^2 }},\quad (x,y) \not= (0,0) \\
1,\quad \quad (x,y) = (0,0) \\
\end{array} \right.\).

Would that be continuous?

 

[Erin0702]

I think it would be discontinuous because f(0,0) does not equal 1...it's undefined when you plug in 0 for x and y

 

[pka]

I think it would be discontinuous because f(0,0) does not equal 1...it's undefined when you plug in 0 for x and y

And you would be correct!
But please don't into a habit of thinking you plug in 0 for x and y because you do not. You exam the behavior of the function in neighborhoods of the point in question.

 

[Erin0702]

Great...thank you so much for the help!