# Distance function: continuous?

• April 5th 2009, 03:38 PM
Andreamet
Distance function: continuous?
Let (X,d) be a metric space. Prove that the distance function
$d: X \times X \rightarrow \mathbb{R}$ is continuous, assuming that XxX has the product topology that results from each copy of X having the topology induced by d.
• April 5th 2009, 04:40 PM
Mush
Quote:

Originally Posted by Andreamet
Let (X,d) be a metric space. Prove that the distance function
$d: X \times X \rightarrow \mathbb{R}$ 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 $f:R \rightarrow R$ is said to be continuous if, at the point $a \in R$, if given $\epsilon > 0$, there is a $\delta > 0$ such that

$|f(x) - f(a)| < \epsilon$

whenever:

$|x-a| < \delta$