Gruber P. Convex and Discrete Geometry

4 Convex Functions
x
epi f
f
C y
Fig. 1.3. Epigraph of a strictly convex function
1.2 Continuity, Support and Differentiability
Using tools from real analysis, continuity, affine support and differentiability properties
of convex functions of one variable were investigated thoroughly in the early
twentieth century. It is difficult to make precise attributions.
In the following we present basic pertinent results. It turns out that a convex
function which is continuous or differentiable, is so in a particularly strong sense.
Let I and J denote (bounded or unbounded) intervals in R. Byint, cl and bd we
mean interior, closure and boundary.
A Simple Preparatory Result
We start with the following simple, yet useful result (Fig. 1.4).
Lemma 1.1. Let f : I → R be convex. Then
f (y) − f (x)
y − x
≤
f (z) − f (x)
z − x
≤
f (z) − f (y)
z − y
for x, y, z ∈ I, x < y < z.
Proof. Let x, y, z ∈ I , x < y < z, and choose 0