Is the Function F(x, y) Continuous at All Points?

[tomboi03]

Let F: R x R -> R be defined by the equation
F(x x y) = { xy/(x^2 + y^2) if x x y \neq 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!

 

[CompuChip]

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
\lim_{x \to 0} F(x, 0) \neq \lim_{y \to 0} F(0, y)
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 :)

 

[Bacle]

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.

 

[Bacle]

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