Solving 2 Questions on Continuity in R^2

[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!

 

[meganw]

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.

 

[Architeuthis Dux]

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!