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.