Example 3

The function $|x|$ is convex on $\mathbf{R}$, since

for $0 \leq \lambda \leq 1$.

It is clear that if $f$ is convex on I, then its restriction to any interval $J \subset I$ is convex on J.

Let $f$ be a convex function on I, and $x, x'$ two points of I such that $x < x'$; if $z \in I$ is exterior to $[x, x']$ then $M_z$ lies above the line D joining $M_x$ and $M_{x'}$; this is an immediate consequence of the lemma.

One deduces from this that if $z$ is a point such that $x < z < x'$ and such that $M_z$ lies on the segment $M_x M_{x'}$, then, for every other point $z'$ such that $x < z' < x'$ the point $M_{z'}$ also lies on the segment $M_x M_{x'}$, for it follows from the above that $M_{z'}$ is at the same time both above and below this segment; in other words, $f$ is then equal to an affine linear function on $[x, x']$.