Can someone help me with 7F from Willard? In part two :
$\mathbf{7}$F. Functions to and from the plane.
The facts presented here for the plane will be proved in more generality for product spaces in Section $8$.
If $f$ is a function on any space $X$ to the plane, associated with $f$ we have the coordinate functions $f_1$ and $f_2$, each mapping $X$ to $\Bbb R$. For each $x\in X$, $f_1(x)$ and $f_2(x)$ are the first and second coordinates, respectively, of $f(x)$.
On the other hand, if $g$ is a function from the plane to any space $Y$, for each fixed $x_0\in\Bbb R$ we can define a function $g_{x_0}$ from $\Bbb R$ to $Y$ by $g_{x_0}(y)=g(x_0,y)$. Similarly, if $y_0\in\Bbb R$ is fixed, $h_{y_0}(x)=g(x,y_0)$ defines a function $h_{y_0}$ from $\Bbb R$ to $Y$. We say $g$ is continuous in $x$ iff $h_{y_0}$ is continuous for each $y_0\in\Bbb R$ and $g$ is continuous in $y$ iff $g_{x_0}$ is continuous for each $x_0\in\Bbb R$.
$2$. If $g:\Bbb R^2\to Y$ is continuous, then it is continuous in both $x$ and $y$.
From S. Willard, General Topology; can anyone help?
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$f$ is continuous in $x$, iff for all fixed $y$, the function $x\mapsto f(x,y)$ is continuous. But this is just $f$ restricted to $X \times\{y\}$, which is continuous as a restriction of a continuous map. Similarly for continuity in $y$.
Is this short proof useful ?
HINT: For any fixed $y_0$, can you write $h_{y_0}(x) = g(x,y_0)$ as a composition of continuous functions?