Let (X,d) be a metric space. Prove that the distance function
$\displaystyle d: X \times X \rightarrow \mathbb{R} $ is continuous, assuming that XxX has the product topology that results from each copy of X having the topology induced by d.
Let (X,d) be a metric space. Prove that the distance function
$\displaystyle d: X \times X \rightarrow \mathbb{R} $ is continuous, assuming that XxX has the product topology that results from each copy of X having the topology induced by d.
Remember, a function $\displaystyle f:R \rightarrow R $ is said to be continuous if, at the point $\displaystyle a \in R$, if given $\displaystyle \epsilon > 0 $, there is a $\displaystyle \delta > 0 $ such that
$\displaystyle |f(x) - f(a)| < \epsilon $
whenever:
$\displaystyle |x-a| < \delta $