Let \(F: X \times Y \to Z\). We say that \(F\) is continuous in each variable separately if for each \(y_0\) in \(Y\), the map \(h: X \to Z\) defined by \(h(x) = F(x \times y_0)\) is continuous, and for each \(x_0\) in \(X\), the map \(k: Y \to Z\) defined by \(k(y) = F(x_0 \times y)\) is continuous. Show that if \(F\) is continuous, then \(F\) is continuous in each variable separately.
Solution
For \(y_0\in Y\) let \(i_{y_0}\colon X\to X\times Y\), \(i_{y_0}(x)=(x,y_0)\), and for \(x_0\in X\) let \(j_{x_0}\colon Y\to X\times Y\), \(j_{x_0}(y)=(x_0,y)\).
1. The maps \(i_{y_0}\) and \(j_{x_0}\) are continuous.
A map into \(X\times Y\) is continuous if and only if its compositions with the projections \(\pi_X\) and \(\pi_Y\) are continuous. Here \(\pi_X\circ i_{y_0}=\operatorname{id}_X\) and \(\pi_Y\circ i_{y_0}\) is the constant map at \(y_0\); likewise \(\pi_X\circ j_{x_0}\) is constant at \(x_0\) and \(\pi_Y\circ j_{x_0}=\operatorname{id}_Y\). Identity and constant maps are continuous.
2. Q.E.D.
The maps of the problem are \(h=F\circ i_{y_0}\) and \(k=F\circ j_{x_0}\), composites of continuous maps by step 1 and the hypothesis on \(F\).