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Advanced Multivariable Calculus / Continuity / Type-o?

  1. Mar 29, 2010 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant 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.

    3. 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?
    Last edited: Mar 29, 2010
  2. jcsd
  3. Mar 29, 2010 #2


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    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.
  4. Mar 29, 2010 #3
    I know. And if I change to polar coordinates it's easy to come up with a limiting value of 0.
  5. Mar 29, 2010 #4


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    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?
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