Functions of two variables, continuity

In summary, the given conversation discusses three functions defined on the x-y plane and their continuity. The functions are f1(x) = (x-y)/(x+y), f2(x,y) = (xy)/(x^2 + y^2), and f3(x,y) = (x^3 - y^3)/(x^2 + y^2). The definition of continuity for a single variable is given and it is assumed to be the same for two variables. To determine the continuity of these functions, the limit as (x,y) approaches (a,b) needs to be evaluated. To prove a function is continuous, the epsilon-delta definition of limit should be used.
  • #1
jbarbarez
1
0
Consider the following functions each of which is defined on the x - y plane

f1(x) = (x-y)/(x+y) if x + y is not 0 and otherwise f1(x,y) = 0

f2(x,y) = (xy)/(x^2 + y^2) if (x,y) is not (0,0) and otherwise f2(0,0) = 0

f3(x,y) = (x^3 - y^3)/(x^2 + y^2) if (x,y) is not (0,0), and otherwise f3(0,0) is 0

Which of these is continuous

A) none B) f1 only C) f2 only D) f3 only E) all three

I know the defination of continuity for a single variable is the lim as x-> a of f(x) = f(a)

So i assume for two variables it should be the lim as (x,y) -> (a,b) of f(x,y) = f(a,b)

But I am not sure how to figure this out can someone help out please
 
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  • #2
jbarbarez said:
I know the defination of continuity for a single variable is the lim as x-> a of f(x) = f(a)

So i assume for two variables it should be the lim as (x,y) -> (a,b) of f(x,y) = f(a,b)

You have the correct definition: if [tex]U[/tex] is an open set in [tex]\mathbb{R}^2[/tex] and [tex](a, b) \in U[/tex], then [tex]f: U \to \mathbb{R}[/tex] is continuous at [tex](a, b)[/tex] if [tex]\lim_{(x, y) \to (a, b)} f(x, y)[/tex] exists and equals [tex]f(a, b)[/tex].

A good way to detect that a function of two real variables is not continuous at a point is to find the limit as you approach the point along lines of different slopes. Often the behavior of the function will be different along different lines.

To prove that a function is continuous at a point, if you don't have another handle on it (such as, the function is built using arithmetic operations and composition on functions that are known to be continuous), start with the epsilon-delta definition of limit.
 

1. What is a function of two variables?

A function of two variables is a mathematical relationship between two variables, where each input value of one variable corresponds to exactly one output value of the other variable. It can be represented in the form of f(x,y) = z, where x and y are the independent variables and z is the dependent variable.

2. How is continuity defined for a function of two variables?

Continuity for a function of two variables is defined similarly to that of a single variable function. A function f(x,y) is continuous at a point (a,b) if the limit of f(x,y) as (x,y) approaches (a,b) exists and is equal to f(a,b). This means that the function has no breaks or gaps at that point and can be drawn without lifting the pen or pencil.

3. How is continuity tested for a function of two variables?

Similar to a single variable function, continuity for a function of two variables can be tested using the following criteria:

  • The function must be defined at the point (a,b).
  • The limit of the function as (x,y) approaches (a,b) must exist.
  • The limit must be equal to the value of the function at (a,b).

4. What is the difference between continuity and differentiability for a function of two variables?

Continuity and differentiability are two related but distinct concepts. A function is continuous if it has no breaks or gaps, while a function is differentiable if it has a well-defined slope at each point. A function can be continuous but not differentiable, but it cannot be differentiable without being continuous.

5. How does the concept of continuity apply to real-world situations?

The concept of continuity applies to real-world situations in many ways. For example, in physics, the continuity of a function represents the smoothness of a physical process or motion. In economics, the continuity of a function can represent the stability of a market or the consistency of a trend. In general, continuity helps us understand the behavior and relationships between different variables in real-world scenarios.

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