Advanced Multivariable Calculus / Continuity / Type-o?

[Jamin2112]

Homework Statement

I don't need to state the whole problem; it's the definitions at the beginning that are giving me trouble.

Homework Equations

So it says,

Definition: A function f(x,y) is continuous at a point (x₀,y₀) if f(x,y) is defined at (x₀,y₀), and if lim_{(x,y)-->(x0,y0)} f(x,y)=f(x₀,y₀).

Definition: A function f(x,y) is discontinuous at a point (x₀,y₀) if it is defined at (x₀,y₀), and if either f(x,y) had no limiting value at (x₀,y₀), or if lim _{(x,y)-->(x0,y0)} f(x,y) has no value.

The problem then gives me a function f(x,y)=(xy²-y³)/(x²+y²) and asks whether lim _{(x,y)-->(0,0)} f(x,y) has a value.

The Attempt at a Solution

Something seems wrong about the definitions. Both of them say that f is defined at (x₀,y₀). But what if f isn't defined there? In the function that I'm given, plugging in 0 for x and 0 for y means diving by zero. Type-o?

 

[Dick]

If f(x0,y0) is undefined, then f(x,y) is discontinuous at (x0,y0). Your book may have a problem with it's phrasing of the definition of 'discontinuous'. But that doesn't mean the limit doesn't exist.

 

[Jamin2112]

But that doesn't mean the limit doesn't exist.

I know. And if I change to polar coordinates it's easy to come up with a limiting value of 0.

 

[Dick]

I know. And if I change to polar coordinates it's easy to come up with a limiting value of 0.

Great. That's completely correct. The limit is zero. But the function is discontinuous unless they choose to define f(0,0)=0. So we agree, right?