# Solving 2 Questions on Continuity in R^2

• Ka Yan
In summary, the conversation is about two questions that need help. The first question asks for a proof of the existence of corresponding real continuous functions given a function defined on a closed subset. The second question asks for a proof that one function is bounded and the other is unbounded in a specific neighborhood. The person responding is unsure about the questions and suggests showing an attempt at solving the problems or using a graphing calculator.
Ka Yan
Two questions need helps

I got two questions below need helps:

1. Let f be a real continuous function defined on a closed subset E of R$$^1$$, then how can I prove the existence of some corressponding real continuous functions g on R$$^1$$, such that g(x)=f(x) for all x$$\in$$E ?

2. Let f and g two functions defined on R$$^2$$ by: f(0,0)=g(0,0)=0, f(x,y)=xy$$^2$$/(x$$^2$$+y$$^4$$), and g(x,y)=xy$$^2$$/(x$$^2$$+y$$^6$$), if (x,y)$$\neq$$(0,0). Then how can I prove that: (1) f is bounded on R$$^2$$, and (2) g is unbounded in every neighborbood of (0,0) ?

Thks!

Last edited:
You're supposed to show some sort of attempt at solving the problem in order to get here. ;)

The question seems a little unclear to me-basically #1 is just asking you to prove that two equations are equal to each other. On #2, You can prove that f is bounded by setting the equations equal to zero and solving for when x=0. Or you can use your graphing calculator.

I could be wrong, (just a high school student), but from the limited information here, that's about all I can come up with.

I can provide some guidance and suggestions for solving these two questions on continuity in R^2.

For the first question, you can use the fact that a continuous function on a closed interval is uniformly continuous. This means that for any ε>0, there exists a δ>0 such that for all x,y in the interval, |f(x)-f(y)|<ε. Using this, you can construct a sequence of functions g_n(x) such that g_n(x) converges to f(x) uniformly on E. This sequence of functions will be continuous since each g_n(x) is constructed by taking the average of f(x) over smaller and smaller intervals. This will prove the existence of a continuous function g(x) on R^1 such that g(x)=f(x) for all x∈E.

For the second question, we can first show that f is bounded on R^2 by considering the limit of f as (x,y) approaches (0,0). Using the squeeze theorem, we can show that the limit is equal to 0, thus f is bounded on R^2.

To prove that g is unbounded in every neighborhood of (0,0), we can choose a sequence of points (x_n,y_n) such that (x_n,y_n) approaches (0,0) and g(x_n,y_n) approaches infinity. This can be done by choosing y_n to be smaller and smaller compared to x_n, such that the denominator of g(x_n,y_n) becomes smaller and smaller, leading to a larger and larger value for g(x_n,y_n). This shows that g is unbounded in every neighborhood of (0,0).

I hope this helps in solving the questions. Remember to always use the definitions and properties of continuity to guide your proofs and constructions. Good luck!

## 1. What is continuity in R^2?

Continuity in R^2 refers to the property of a function where the graph of the function can be drawn without any breaks or holes. This means that the function is well-behaved and has a smooth, continuous curve without any abrupt changes in direction or value.

## 2. How do you determine if a function is continuous in R^2?

To determine if a function is continuous in R^2, you need to check three conditions: 1) the function is defined at a given point, 2) the limit of the function exists at that point, and 3) the limit is equal to the value of the function at that point. If all three conditions are satisfied, then the function is continuous at that point.

## 3. What is the difference between continuity and differentiability in R^2?

Continuity and differentiability are two related but distinct concepts in math. Continuity refers to the smoothness of a function's graph, while differentiability refers to the ability to find a derivative at a point. A function can be continuous but not differentiable, but it cannot be differentiable without being continuous.

## 4. What are the common types of discontinuities in R^2?

There are three common types of discontinuities in R^2: removable, jump, and infinite. A removable discontinuity occurs when there is a hole in the graph of the function, a jump discontinuity occurs when there is a sudden change in the function's value, and an infinite discontinuity occurs when the function approaches infinity at a certain point.

## 5. How can the concept of continuity in R^2 be applied in real-life situations?

The concept of continuity in R^2 has various applications in real-life situations, such as in physics, engineering, and economics. For example, the continuity of a function can be used to model the smoothness of motion in physics, the stability of structures in engineering, and the market demand and supply in economics. It can also be used to optimize functions and make accurate predictions in various fields.

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