I was studying multivariable calculus and I cam accross a theorem recently. I am having some trouble understanding the theorem's wording. The theorem is:
Let $u=f(x,y)$ and $v=g(x,y)$ be both defined in the same region $R$ of the $xy$ plane. Let $\lim_{(x,y)\to(a,b)}f(x,y)=A$ and and $\lim_{(x,y)\to(a,b)}g(x,y)=B.$ Let $F(u,v)$ be defined and continuous in a region $R_0$ of the $uv$ plane and let $F(f(x,y),g(x,y))$ be defined for $(x,y)$ in $R.$ Then, if $(A, B)$ is in $R_0,$ $$\lim_{(x,y)\to(a,b)}F(f(x,y),g(x,y))=F(A,B).$$ If further,$f(x,y)$ and $g(x,y)$ are continuous at $(a,b)$ so also is $F(f(x,y),g(x,y))$.
First of all, I don't get what is meant by "$F(u,v)$ be defined and continuous in a region $R_0$ of the $uv$ plane". Specifically, I have no idea what is meant by the $uv$ plane. Any clarification regarding this statement will be greatly appreciated.
Secondly, the line "let $F(f(x,y),g(x,y))$ be defined for $(x,y)$ in $R$" seems to suggest that there is a particular point say, $(x,y)$ in $R$ that is also in the range of definition of $F$, i.e $R_0.$ I am unsure whether this interpretation is correct.
The $uv$ plane is just the Cartesian plane $\mathbb{R}^2$, but instead of being indexed by $(x, y)$ points it's indexed by $(u, v)$ points.
The idea being presented here is:
You have two functions $f(x, y)$ and $g(x, y)$ that are both defined in some region $R$ of $\mathbb{R}^2$.
You have another function $F(u, v)$ that is defined in some other region $R_0$ of $\mathbb{R}^2$.
When you take $(x, y) \in R$, the point $(f(x, y), g(x, y))$ belongs to $R_0$ - in other words, setting $u = f(x, y)$ and $v = g(x, y)$ gives you a point in the region where $F(u, v)$ is defined.
As $(x, y)$ approaches the limit point $(x_0, y_0)$, $f(x, y) \rightarrow A$ and $g(x, y) \rightarrow B$.
As a result, you can then say that as $(x, y) \rightarrow (x_0, y_0)$, you also have $F(f(x, y), g(x, y)) \rightarrow F(A, B)$.
Or in other words, if you compose continuous functions, the result is also continuous.