Distance Function of Metric Space is Continuous

Theorem
Let $\left({M, d}\right)$ be a metric space.

Then for all $x, y \in M$, the mapping $\xi \mapsto d \left( {\xi, y} \right)$ is continuous in $x$.

Proof
Let $x_0, y \in M$.

Let $x \in M$ be such that $d \left( {x, x_0} \right) < \epsilon$.

Then by the reverse triangle inequality:
 * $\left \vert{d \left({x, y}\right) – d \left({x_0, y}\right)}\right \vert \le d \left({x, x_0}\right) < \epsilon$

Let $\delta = \epsilon$.

The result follows by the Metric Space Continuity by Epsilon-Delta.