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!
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 :)
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.
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