L1 Metric on Closed Real Interval is Metric

Theorem
Let $S$ be the set of all real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let the real-valued function $d: X \times X \to \R$ be defined as:
 * $\displaystyle \forall f, g \in X: d \left({f, g}\right) := \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t$

Then $\left({X, d}\right)$ is a metric space.