Continuity of $f(x,y)=x$ in Product Metric

Question

Let $(X \times Y,d)$ be the product metric space of $(X,d_1)$ and $(Y,d_2)$. Show that the function goes to $X$ from $X \times Y$ defined by $f(x,y)=x$ is continuous.

I really have any ideas about how to prove it. How I can show this?

Answer 1

f:X×Y -> X×Y, (x,y) -> (x,y), the identity map, is continuous.
Restrict f to X×{y} to get a function with f(x,y) = x, for all x in X.
Finally show any constriction of a continuous function is continuous.

Answer 2

You can use the $\epsilon-\delta$ criterion for continuity. Let $(x_0,y_0) \in X \times Y$ and let $\epsilon > 0$. You can take $\delta =\epsilon $ and then if $d((x,y),(x_0,y_0)) = max \left\{ d_1(x,x_0),d_2(y,y_0) \right\} < \delta = \epsilon$ we have $d_1(f(x,y),f(x_0,y_0))=d_1(x,x_0) < \epsilon $.