Showing F is not continuous

  1. Let F: R x R -> R be defined by the equation
    F(x x y) = { xy/(x^2 + y^2) if x x y [tex]\neq[/tex] 0 x 0 ; 0 if x x y = 0 x 0
    a. Show that F is continuous in each variable separately.
    b. Compute the function g: R-> R defined by g(x) = F(x x x)
    c. Show that F is not continuous.

    I know how to do part a....
    but I'm not sure how to do b or c.

    If you can help me out that would be great! thank you!
     
  2. jcsd
  3. CompuChip

    CompuChip 4,299
    Science Advisor
    Homework Helper

    Well, b seems rather straightforward, just plug it in.

    For c, you could show that there is a point for which the limit value depends on the path you take. For example, showing that
    [tex]\lim_{x \to 0} F(x, 0) \neq \lim_{y \to 0} F(0, y)[/tex]
    would prove that F is not continuous at (0, 0) because then it shouldn't matter how you get to (0, 0). I think that b should give you a hint on which point and paths to consider :)
     
  4. Sorry to rehash something so old; I was doing a search for the general situation;
    wonder if someone knows the answer:

    An important/interesting question would be if we can add some new condition
    so that if f(x,.) and f(.,y) are continuous, then so is f(x,y).

    For one thing, the continuity of maps f:XxY-->Z is often used in constructing
    homotopies; I have never seen the issue of why/when these homotopies are
    continuous.
     
  5. Sorry, I can't access the 'Edit' button for some reason.

    A standard counter to having a function beeing continuous in each variable, yet
    not overall continuous is the one given by tomboi03.

    My point is that a homotopy between functions f,g, is defined to be a _continuous_ map H(x,t) with H(x,0)=f and H(x,1)=g. Since we cannot count on H(x,t) being continuous when each of H(x,.) and H(.,y) is continuous :what kind of result do we use to show that our map H(x,t) is continuous? Do we use the 'good-old' inverse image of an open set is open , or do we use the sequential continuity result that [{x_n}->x ] ->[f(x_n)=f(x)]
    (with nets if necessary, i.e., if XxI is not 1st-countable)?

    I saw a while back an interesting argument that if continuity on each variable alone
    was enough to guarantee continuity, then every space would have trivial fundamental group:

    e.g, for S^1, use H(e^i*2Pi*t,s) :=e^i2Pi(t)^s
     
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