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About continuity.

  1. Mar 9, 2008 #1
    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[tex]^1[/tex], then how can I prove the existence of some corressponding real continuous functions g on R[tex]^1[/tex], such that g(x)=f(x) for all x[tex]\in[/tex]E ?

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

    Thks!
     
    Last edited: Mar 10, 2008
  2. jcsd
  3. Mar 16, 2008 #2
    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, thats about all I can come up with.
     
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