For what domains is this function continuous for?

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SUMMARY

The function f(x,y) = 1/(x^2 + y^2 - 1) is continuous for all domains except where the denominator equals zero, specifically at the points (x, ±√(1 - x^2)) for any real number x. For the function to be classified as C^1, meaning that its first derivatives are continuous, it is also not defined at the same points, confirming that f is C^1 for the domain {(x,y) | x^2 + y^2 ≠ 1}. The analysis confirms that the function's continuity and differentiability are directly linked to the behavior of its denominator.

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  • Understanding of multivariable calculus concepts, specifically continuity and differentiability.
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  • Study the concept of continuity in multivariable functions, focusing on the implications of denominators approaching zero.
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Jenny010
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Homework Statement



f(x,y) = 1/(x^2 + y^2 -1)

1. For what domains is f continuous?
2. For what domains is f a C^1 function? (Here C^1 means that the first derivatives of f are all continuous)


Homework Equations





The Attempt at a Solution



I would be very grateful for the help you can give me with this, because I'm just not sure if I'm coming to the right conclusions or if there is something I'm missing. If you could also keep to relatively simple maths as I haven't done a lot before (keeping to simple explanations or methods would be so helpful).

For the first part I believe that f would be continuous when the denominator is not zero. So what I get for the first part is that it is continuous for all domains apart from:
(x , + or - SQRT(1 - x^2)) where x is any real number

For the second part I get the following two first derivatives:
df/dx = (-2x)/((x^2 + y^2 - 1)^2)
df/dy = (-2y)/((x^2 + y^2 - 1)^2)
And from this I get that f is C^1 for:
(x , + or - SQRT(1 - x^2)) where x is any real number

Can anyone give me some help with this? I'm not sure if I am on the right track with my answers or if I have missed some domains for which it is not or is continuous/C^1 for.

Thank you for helping,

Jenny
 
Physics news on Phys.org
You got it. You may want to give the domain as {(x,y) | x^2+y^2[itex]\neq[/itex]1 }.
 

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