
#1
Feb2409, 01:26 AM

P: 77

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
Feb2409, 03:08 AM

Sci Advisor
HW Helper
P: 4,301

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 :) 



#3
Jul1311, 04:25 PM

P: 662

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. 



#4
Jul1411, 12:52 PM

P: 662

showing F is not continuous
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 'goodold' 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 1stcountable)? 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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