Absolute Value Function is Convex/Proof 1

Proof
Let $x_1, x_2, x_3 \in \R$ such that $x_1 < x_2 < x_3$.

Consider the expressions:
 * $\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1}$
 * $\dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$

The following cases are investigated:


 * $(1): \quad x_1, x_2, x_3 < 0$:

Then:


 * $(2): \quad x_1, x_2, x_3 > 0$:

Then:


 * $(3): \quad x_1 < 0, x_2, x_3 > 0$:


 * $(4): \quad x_1, x_2 < 0, x_3 > 0$:

Thus for all cases, the condition for $f$ to be convex is fulfilled.

Hence the result.