Say $X$, $Y_1$ and $Y_2$ are topological spaces. Let $f_1 \; X \to Y_1$ and $f_2 \; X \to Y_2$.
If $f\; X \to Y_1 \times Y_2 $ $f(x) = (f_1(x), f_2(x))$
$Y_1 \times Y_2$ is a topological space with the product topology.
How do I prove that $f$ is continuous in $x$ if and only if $f_1$ and $f_2$ are continuous in $x$ ?