Gruber P. Convex and Discrete Geometry

1 Convex Functions of One Variable 11
The above definition of second-order differentiability is slightly weaker than
the common definition. In this way it still embodies the main idea of the common
second-order differentiability and has the advantage that it applies to general convex
functions as the following result shows:
Theorem 1.7. Let I be open and f : I → R convex. Then f is twice differentiable
almost everywhere on I . Moreover, for almost every x ∈ I,
f (y) = f (x) + f ′ (x)(y − x) + 1
2 f ′′ (x)(y − x) 2 + o(|y − x| 2 ) as y → x, y ∈ I.
Proof. We first state two well-known theorems of Lebesgue, see [499].
(9) A non-decreasing real function on I is almost everywhere differentiable.
(10) The derivative of an absolutely continuous function g : J → R exists
almost everywhere on J, is Lebesgue integrable and, for x ∈ J,
�y
g(y) = g(x) + g ′ (t) dt for y ∈ J.
By Theorem 1.4 above and (9), there are a countable set M and a set N of measure
0, both in I , such that the following statements hold: f ′ (x) exists and equals
f ′ − (x) for each x ∈ I \M and f ′ − exists on I and is non-decreasing, and f ′′
− (x) exists
for each x ∈ I \N. Thus,
(11) f ′′
− (x) = lim
y→x
= lim
y→x
y∈I \M
x
f ′ − (y) − f ′ − (x)
y − x
f ′ (y) − f ′ (x)
y − x
= lim
y→x
y∈I \M
f ′ − (y) − f ′ − (x)
y − x
= f ′′ (x) for x ∈ I \(M ∪ N),
concluding the proof of the first assertion of the theorem.
To show the second assertion, note that (11) yields the following:
(12) f ′ (y) = f ′ (x) + f ′′ (x)(y − x) + o(|y − x|)
as y → x, y ∈ I \M for x ∈ I \(M ∪ N).
Since by Theorem 1.1 f is absolutely continuous, Proposition (10) shows that one
may integrate (12) from x to y to obtain
f (y) = f (x) + f ′ (x)(y − x) + 1
2 f ′′ (x)(y − x) 2 + o(|y − x| 2 )
as y → x, y ∈ I \M for x ∈ I \(M ∪ N).
Now note that the continuity of both sides of this equality permits us to cancel the
restriction that y �∈ M. ⊓⊔
Theorem 1.5 says that for any convex function f the differentiability of f implies
that f ′ is continuous, i.e. f is of class C 1 . Examples show that there is no analogous
result for f ′′ :If f ′′ exists it need not be continuous.