How to show continuous at each point in R^2

In summary, continuity at a point in R^2 means that a function f(x,y) has no sudden jumps or breaks at that point (a,b) and the limit of the function as (x,y) approaches (a,b) exists and is equal to f(a,b). To prove continuity at a point, the limit must also be equal to the function evaluated at (a,b). The necessary conditions for continuity at a point are that f(a,b) must be defined, the limit must exist, and the limit must be equal to f(a,b). It is possible for a function to be continuous at a specific point in R^2 but not at every point in its domain. This concept is applicable in various real-life situations,
  • #1
misterau
20
0

Homework Statement



f(x , y) = y^3 + x^3

Calculate the partial derivatives fx and fy and show they are continuous at each point (x,y) ∈ R^2

Homework Equations


A function is continuous on a region R in the xy-plane if it is continuous at each point in R

A function f is continuous at the point (a,b) if
lim f(x,y) = f(a,b)
(x,y)->(a,b)


The Attempt at a Solution


Now calculating the partial derivatives was easy. But not sure how to show show they are continuous at each point (x,y) ∈ R^2?
 
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  • #2
There are some theorems which are very useful in cases like this, like
[itex]x \mapsto x[/itex] is continuous
a scalar multiple of a continuous function, as well as the sum of two continuous functions is continuous
a product of two continuous functions is continuous
a quotient of two continuous functions is continuous as long as the denominator is non-zero
the composition of continuous functions is continuous, where defined

There should be such a set of theorems which is proved almost right after the definition of continuous, which we usually apply instead of the definition.

If you want to do the definition, you should calculate the limit, for example
[tex]\lim_{(x, y) \to (a, b)} (3 y^2 + x^3)[/tex]
and show that it is [itex]3b^2 + a^3[/itex].
 
  • #3
CompuChip said:
There are some theorems which are very useful in cases like this, like
[itex]x \mapsto x[/itex] is continuous
a scalar multiple of a continuous function, as well as the sum of two continuous functions is continuous
a product of two continuous functions is continuous
a quotient of two continuous functions is continuous as long as the denominator is non-zero
the composition of continuous functions is continuous, where defined
.
I am not understanding how this helps me do the problem? If you could show me an example or link to an example I would be thankful.
 
  • #4
nvm I worked it out. Thanks for the help.
 
  • #5
OK, here is an example:

Let f(x, y) = x + y2.

By the first theorem (or a variation on it), [itex](x, y) \mapsto x[/itex] and [itex](x, y) \mapsto y[/itex] are continuous. Multiplying the second with itself, [itex](x, y) \mapsto y^2[/itex] is continuous. Using the sum rule on the first and the "y-squared" function, f(x, y) is continuous.
 

1. What does it mean for a function to be continuous at a point in R^2?

A function f(x,y) is continuous at a point (a,b) in R^2 if the limit of f(x,y) as (x,y) approaches (a,b) exists and is equal to f(a,b). In other words, the function has no sudden jumps or breaks at that point.

2. How do you prove continuity at a point in R^2?

To prove continuity at a point (a,b) in R^2, you must show that the limit of the function as (x,y) approaches (a,b) exists and is equal to the function evaluated at (a,b). This can be done using the definition of continuity or by using the properties of continuity.

3. What are the necessary conditions for a function to be continuous at a point in R^2?

The necessary conditions for a function f(x,y) to be continuous at a point (a,b) in R^2 are:

  • f(a,b) must be defined
  • the limit of f(x,y) as (x,y) approaches (a,b) must exist
  • the limit must be equal to f(a,b)

4. Can a function be continuous at a point in R^2 but not at every point?

Yes, it is possible for a function to be continuous at a specific point (a,b) in R^2 but not at every point in its domain. This means that the function may have sudden jumps or breaks at other points, but it is continuous at (a,b).

5. How can continuity at a point in R^2 be applied in real-life situations?

Continuity at a point in R^2 is important in many real-life situations, such as in physics and engineering. For example, the concept of continuous motion is used in the study of fluid dynamics and the design of structures like bridges. In economics, the concept of continuity is used in models to predict changes in supply and demand. In general, continuity at a point helps us understand how a function behaves and can be applied to various fields of study.

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