For what domains is this function continuous for?

In summary, the function f(x,y) = 1/(x^2 + y^2 -1) is continuous for all domains except for (x , + or - SQRT(1 - x^2)), where x is any real number. It is a C^1 function for all domains except for (x , + or - SQRT(1 - x^2)), where x is any real number. This is because the denominator cannot be equal to zero for the function to be continuous or C^1.
  • #1
Jenny010
1
0

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
 
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  • #2
You got it. You may want to give the domain as {(x,y) | x^2+y^2[itex]\neq[/itex]1 }.
 

FAQ: For what domains is this function continuous for?

1. What does it mean for a function to be continuous?

Continuity refers to the property of a function where there are no breaks, gaps, or jumps in the graph. In other words, a continuous function has a smooth and connected graph without any sharp corners or holes.

2. What are the three criteria for continuity?

The three criteria for continuity are: 1) the function must be defined at the point in question, 2) the limit of the function at that point must exist, and 3) the limit must equal the value of the function at that point.

3. What are the different types of continuity?

There are three types of continuity: pointwise continuity, uniform continuity, and differentiable continuity. Pointwise continuity means that the function is continuous at each individual point, uniform continuity means that the function is continuous over an entire interval, and differentiable continuity means that the function is continuous and differentiable at a point.

4. What are the main domains for which a function is continuous?

A function can be continuous over a variety of domains, but the main domains for which continuity is often discussed are intervals, closed intervals, open intervals, and half-open intervals. The specific domain for which a function is continuous will depend on the specific function and its properties.

5. How can I determine if a function is continuous for a given domain?

To determine if a function is continuous for a given domain, you must check if the three criteria for continuity are met. Also, you can use graphical and algebraic methods, such as the Intermediate Value Theorem, to determine continuity over a specific domain. Additionally, you can check if the function has any points of discontinuity, such as vertical asymptotes or removable discontinuities, within the given domain.

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