Continuity of a multivariable function

In summary, the question is whether the given function is continuous. The attempt at a solution involved using various substitutions to try and prove that the limit as (x,y)→(2,0) does not exist, but this approach was unsuccessful. The suggestion is to convert to ellyptic coordinates and use L'hopital's rule to compute the limit as r tends to 0, and if the limit is not identically 0 (or depends on t), then the function is not continuous at the points on the ellipse. Another approach suggested is to substitute in the function and compute the limit as r tends to 1, with t being constant.
  • #1
Jalo
120
0

Homework Statement



Given the function:

x*y / (4-x²-2y²) if x²+2y² ≠4
0 if x²+2y² = 4

Check if the function is continuous.



Homework Equations





The Attempt at a Solution



I tried using various ways to see if the result of the limit as (x,y)→(2,0) was the same, such as y=x-2, y=(x-2)², etc..
I didn't manage to prove that the limit didn't existed. I always arrive at the 0/0 indetermination...

If anyone could point me in the right direction I'd appreciate!

D.
 
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  • #2
convert to ellyptic coordinates x=2rcos(t); y=sqt(2)rsin(t) and compute the limit as r tends to 0,using L'hopital's rule.If the limit is not identically 0 (or depends on t),then the function is not continuous at the poins on the ellipse.
 
  • #3
hedipaldi said:
convert to ellyptic coordinates x=2rcos(t); y=sqt(2)rsin(t) and compute the limit as r tends to 0,using L'hopital's rule.If the limit is not identically 0 (or depends on t),then the function is not continuous at the poins on the ellipse.

Hmm I'll try doing it. However I never learned ellyptic coordinates. I'm wondering if there's any other way to solve this!
 
Last edited:
  • #4
o.k don't name it,just substitute in the function and compute the limit as r tends to 1 and t is constant.
 

FAQ: Continuity of a multivariable function

What is the definition of continuity for a multivariable function?

Continuity of a multivariable function means that the function remains unchanged at a given point when the input variables are changed by a small amount. In other words, the limit of the function as the input variables approach a point must exist and be equal to the value of the function at that point.

How is continuity of a multivariable function different from continuity of a single variable function?

In a single variable function, continuity is determined by the behavior of the function at a single point. However, in a multivariable function, continuity is determined by the behavior of the function at every point in the domain. This means that all possible paths to a point must produce the same limit for the function to be continuous.

Can a multivariable function be continuous at a point but not differentiable?

Yes, it is possible for a multivariable function to be continuous at a point but not differentiable. This can occur when the function has sharp turns or corners at that point. In order for a function to be differentiable, it must also be smooth and have a well-defined tangent plane at that point.

Are all continuous multivariable functions also differentiable?

No, not all continuous multivariable functions are differentiable. A function may be continuous at a point but not differentiable if it has sharp turns or corners at that point. However, if a function is differentiable, it must also be continuous.

How is the continuity of a multivariable function tested?

The continuity of a multivariable function can be tested using the three-variable limit theorem. This theorem states that if a function is continuous at a point, the limit of the function as three variables approach that point must be equal to the value of the function at that point. Additionally, the continuity of a multivariable function can also be checked by evaluating the function along different paths to the point in question and ensuring that the limit is the same for all paths.

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