Distance from Point to Subset is Continuous Function

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

Let $A \subseteq X$ be a non-empty subset of $X$.

Let $f: X \to \R$ be the function defined as:
 * $\forall x \in X: f \left({x}\right) = d \left({x, A}\right)$

where $d \left({x, A}\right)$ denotes the distance from $x$ to $A$.

Then $f$ is continuous.