Show that a function is continuous in one variable.

Question

Let $F: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be defined by the equation $$ F(x,y) = \begin{cases} \dfrac{xy}{(x^2+y^2)} & \textrm{ if } (x,y) \neq (0,0) \\ 0 & \textrm{ if } (x,y) = (0,0). \end{cases} $$ I am trying to show that $F$ is continuous in each variable separately. What does this exactly mean?

Does it mean we have to show that the function $$ F(x) = \dfrac{xy_0}{(x^2+y_0^2)} $$ is continuous for fixed $y_0 \in \mathbb{R}$? Any ideas how to do that?

Answer 1

Yes, that's what you have to do (since $f(x,y)=f(y,x)$). THat's easy. If $y_0\neq0$, you have a rational function, which is continuous. And if $y_0=0$, you have the null function, which is continuous too.

If it was not true that $f(x,y)=f(y,x)$, then you would have to do this for each variable.

Answer 2

For fixed $y_0$, let $g_{y_0}(x)=F(x,y_0)=xy_0/(x^2+y_0^2)$. We are just doing this to be totally clear that $g_{y_0}$ is a real function in one variable, and we know how to talk about the continuity of $g_{y_0}$. Well, $g_{y_0}$ is a ratio of continuous functions, so it is continuous unless $x^2+y_0^2=0$. This is only possible if both $x$ and $y_0$ are $0$, so we need to show that $g_0$ is continuous at $0$. If you write out that $g_0$ is, this is trivial.