Gruber P. Convex and Discrete Geometry

1 Convex Functions of One Variable 9
Let x ∈ I and assume first that f ′ − is continuous at x. By(2), f ′ − (x) ≤ f ′ + (x) ≤
f ′ − (y) for all y ∈ I with x < y. Letting y → x + 0, the continuity of f ′ − at x yields
f ′ − (x) = f ′ + (x) which, in turn, shows that f ′ (x) exists. Assume, second, that f ′ − is
not continuous at x. Then (2) and (5) imply
f ′ − (x) < lim
y→x+0 f ′ − (y) ≤ lim
y→x+0 f ′ + (y) = f ′ + (x).
Hence f ′ (x) does not exist. The proof of (6) is now complete.
The theorem finally follows from (2) and (6) on noting that a non-decreasing
function on I has at most countably many points of discontinuity. ⊓⊔
A less precise extension of Theorem 1.4 to convex functions in d variables is due
to Reidemeister 2.6. A precise extension to convex bodies in E d is the Anderson–
Klee theorem 5.1.
An important consequence of Theorem 1.4 is the following result.
Theorem 1.5. Let I be open and f : I → R convex. If f is differentiable on I , then
f ′ is continuous, i.e. f is of class C 1 .
This result can be extended to d dimensions, compare Theorem 2.8.
First-Order Differentiability and Affine Support
The following result shows that, for convex functions of one variable, the relation
between differentiability and affine support is particularly simple.
Proposition 1.2. Let f : I → R be convex and x ∈ int I . Then an affine function
a : R → R of the form a(y) = f (x) + u(y − x) for y ∈ R is an affine support of f
at x if and only if
f ′ − (x) ≤ u ≤ f ′ + (x).
Proof. Denote the coordinates in E2 by y and z. By Theorem 1.4, the left and right
derivatives f ′ − (x) and f ′ + (x) exist. Their definitions show that the half-lines
(7) z = f (x) + f ′ − (x)(y − x) for y ≤ x,
z = f (x) + f ′ + (x)(y − x) for y ≥ x,
are the left and right half-tangents of the curve z = f (y) at y = x (Fig. 1.5).
As a consequence of Propositions (3) and (4) in the proof of Theorem 1.4 we have
(8) f (y) ≥ f (x) + f ′ − (x)(y − x) for y ∈ I, y ≤ x,
f (y) ≥ f (x) + f ′ + (x)(y − x) for y ∈ I, y ≥ x,
f ′ − (x) ≤ f ′ + (x).