In this note a mean value theorem for continuous vector functions is introduced by mollified derivatives and smooth approximations

La Torre Davide

Rocca Matteo

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
UNIVERSITÀ DELL'INSUBRIA 
FACOLTÀ DI ECONOMIA 
 
http://eco.uninsubria.it 
 
Davide La Torre e Matteo Rocca 
 
Mean value theorem for 
continuous vector functions by 
smooth approximations 
 
2002/12  
© Copyright D. La Torre e M. Rocca 
Printed in Italy in April 2002 
Università degli Studi dell'Insubria 
Via Ravasi 2, 21100 Varese, Italy 
 
All rights reserved. No part of this paper may be reproduced in 
any form without permission of the Author. 
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Faculty of the University of Insubria. Mean value theorem for continuous vector
functions by smooth approximations
Davide La Torre∗ Matteo Rocca†
Abstract
In this note a mean value theorem for continuous vector functions is in-
troduced by molliﬁed derivatives and smooth approximations.
1 Preliminary deﬁnitions
In this paper, a generalized mean value theorem for continuous vector functions is
proved. This result involves generalized derivatives, deﬁned by smooth approxima-
tions, following the approach introduced by Craven and Ermoliev, Norkin, Wets
([3, 4]). In particular, when local lipschitzianity is assumed, our mean value theo-
rem reduces to the well known mean value theorem expressed by means of Clarke’s
generalized Jacobian [2].
We will make use of the following classical deﬁnitions and results of Functional
Analysis.
Deﬁnition 1.1. A sequence of molliﬁers is any sequence of functions {φ} : Rn →
R+,  ↓ 0, such that:
• suppφ := {x ∈ Rn,| φ(x) > 0} ⊆ ρclB, ρ ↓ 0,
•
R
Rn φ(x)dx = 1,
∗University of Milan, Department of Economics, Faculty of Political Sciences, via Conservato-
rio,7, 20122, Milano, Italy. e-mail: davide.latorre@unimi.it
†University of Insubria, Department of Economics, Faculty of Economics, via Ravasi, 2, 21100,
Varese, Italy. e-mail: mrocca@eco.uninsubria.it
1where B is the unit ball in Rn, clX means the closure of the set X and dx denotes
Lebesgue measure.
Example 1.1. [4] Let  be a positive number.
(i) The functions:
φ(x) =

 
 
1
n, max1,...,n |xi| ≤ 
2
0, otherwise
are called Steklov molliﬁers.
(ii) The functions:
φ(x) =

  
  
C
n exp

2
kxk2−2

, kxk < 
0, kxk ≥ 
with C ∈ R such that
R
Rn φ(x)dx = 1, are called standard molliﬁers.
It is easy to check that the second family of functions is smooth.
Deﬁnition 1.2. [4] Given a locally integrable function f : Rn → Rm and a sequence
of bounded molliﬁers, deﬁne the functions f(x) through the convolution:
f(x) :=
Z
Rn
f(x − z)φ(z)dz =
Z
Rn
f(z)φ(x − z)dz.
The sequence f(x) is said a sequence of molliﬁed functions.
Remark 1.1. There is no loss of generality in considering f : Rn → Rm. The
results in this paper remain true also if f is deﬁned on an open subset of Rn.
Proposition 1.1. [4] Let f ∈ C (Rn). Then f converges continuously to f, i.e.
f(x) → f(x) for all x → x. In fact f converges uniformly to f on every compact
subset of Rn as  ↓ 0.
Molliﬁed functions have also some diﬀerentiability properties, under suitable
regularity assumptions on f and the associated molliﬁers, as stated in the following:
Proposition 1.2. [5] Let f : Rn → Rm be locally integrable. Whenever the molliﬁers
φ are of class Ck, so are the associated molliﬁed functions. Furthermore if φ are
of class Ck,1, that is k-times diﬀerentiable with locally lipschitzian Jacobians, then
so are the associated molliﬁed functions.
2By means of molliﬁed functions it is possible to deﬁne generalized directional
derivatives for a nonsmooth function f. Such an approach has been deepened by
several authors (see e.g. [3, 4]) in the scalar case.
Deﬁnition 1.3. Let f : Rn → Rm be a locally integrable function, let n ↓ 0 as
n → +∞ and consider the sequence fn := fn of molliﬁed functions with associated
molliﬁers φn ∈ C1. Given x,d ∈ Rn we deﬁne the following sets:
∂f(x0;d) = {l = lim
n→+∞∇fn(xn)d,xn → x0}
∂∞f(x0;d) = {l = lim
n→+∞tn∇fn(xn)d,xn → x0,tn ↓ 0
+}\{0}.
Proposition 1.3.
• ∂f(x0;d) is a closed subset of Rm.
• ∂∞f(x0;d) is a closed cone of Rm.
• ξ∂f(x0;d) ⊆ ∂(ξf)(x0;d), ∀ξ ∈ Rm. If f is locally lipschitzian then the equality
holds.
Proof. Omitted since trivial.
Proposition 1.4. If f is locally lipschitzian then ∂f(x0;d) ⊆ ∂Cf(x0)d, where
∂Cf(x0) is Clarke’s generalized Jacobian of f at x0 [2].
Proof. In fact, ∀ξ ∈ Rm, the following inclusion holds [4]:
∂(ξf)(x0;d) ⊆ ∂C(ξf)(x0)d.
Hence:
ξ∂f(x0;d) ⊆ ξ∂Cf(x0)d
and then the thesis follows by a standard separation argument.
Corollary 1.1. If f is C1 then ∂f(x0;d) = ∇f(x0)d.
Proof. If f is C1, then ∂Cf(x0)d = ∇f(x0)d [2] and then the thesis follows from the
previous proposition.
32 Generalized mean value theorem
Theorem 2.1. Let f : Rn → Rm be a given continuous function. Then the following
mean value theorem holds:
f(x) − f(y) ∈

conv δ∈[x,y]∂f(δ;x − y) + conv δ∈[x,y]∂∞f(δ;x − y) ∪ {0}
	[
conv δ∈[x,y] {∂∞f(δ;x − y) + f(x) − f(y)}.
where conv δ∈[x,y]A(δ) denotes the convex hull of the sets A(δ), δ ∈ [x,y].
Proof. In fact, for the scalar function ξfn, we have:
ξfn(x) − ξfn(y) = ξ∇fn(δn(ξ))(x − y)
where ξ ∈ Rm and δn(ξ) ∈ (x,y). So we have:
ξfn(x) − ξfn(y) ∈ ξAn
where An = {∇fn(δ)(x − y),δ ∈ [x,y]} and obviously An is compact. So by a
standard separation argument, we have:
fn(x) − fn(y) ∈ convAn
where conv stands for the convex hull of An. Let now ln = fn(x) − fn(y). For all
n ∈ N, by Charatheodory theorem, we have:
ln =
m+1 X
j=1
λj,naj,n,
where
Pm+1
j=1 λj,n = 1, λj,n ≥ 0, j = 1,...,m + 1, aj,n ∈ An. Then:
ln =
X
j∈I1
λj,naj,n +
X
j∈I2
λj,naj,n +
X
j∈I3
λj,naj,n
where:
• for all j ∈ I1 the sequence aj,n is bounded and it converges to aj,0. Since
aj,n ∈ An, ∀n ∈ N, then aj,n = ∇fn(δj,n)(x − y), δj,n ∈ [x,y]. Eventually by
extracting a subsequence, we have δj,n → δj ∈ [x,y] and then:
aj,0 = lim
n→+∞aj,n = lim
n→+∞∇fn(δj,n)(x − y) ∈ ∂f(δj;d).
4• for all j ∈ I2, the sequence aj,n is unbounded but the sequence λj,naj,n is
bounded and it converges to aj,∗.
• for all j ∈ I3, the sequence λj,naj,n is unbounded but there exists j0 ∈ I3 such
that the sequence
λj,naj,n
kλj0,naj0,nk converges to aj,∞, ∀j = 1,...,m + 1.
We now consider the case in which I3 is not empty. Then:
0 = lim
n→+∞
ln
kλj0,naj0,nk
=
lim
n→+∞
X
j∈I3
λj,naj,n
kλj0,naj0,nk
=
X
j∈ I3
aj,∞,
with aj0,∞ 6= 0. Since aj,n = ∇fn(δj,n)(x − y), δj,n → δj,
λj,n
kλj0,naj0,nk → 0 for every
j ∈ I3, we have aj,∞ ∈ ∂∞f(δj;d) ∪ {0}. Furthermore aj0,∞ 6= 0 and then:
0 ∈ conv δ∈[x,y]∂∞f(δ;x − y).
We now consider the case in which I3 is empty. Eventually extracting subsequences,
let λj,0 = limn→+∞ λj,n. Then, we have λj,0 = 0 ∀j ∈ I2,
P
j∈I1 λj,0 = 1 and
aj,∗ ∈ ∂∞f(δj;x − y) ∪ {0}. So:
l = lim
n→+∞ln =
X
j∈I1
λj,0aj,0 +
X
j∈I2
aj,∗
Obviously
P
j∈I2 aj,∗ ∈ conv δ∈[x,y]∂∞f(δ,x − y) ∪ {0}. So we have:
f(x) − f(y) ∈

conv δ∈[x,y]∂f(δ,x − y) + conv δ∈[x,y]∂∞f(δ,x − y) ∪ {0}
	[
conv δ∈[x,y] {∂∞f(δ,x − y) + f(x) − f(y)}.
Corollary 2.1. Let f : Rn → R. If we deﬁne a generalized upper derivative as:
Df(x;d) = limsup
n→+∞,xn→x0
∇fn(xn)d
then the following mean value theorem holds:
f(x) − f(y) ≤ Df(ξ;x − y)
where ξ ∈ [x,y].
5Proof. We only consider the case in which Df(s,x − y) < +∞ ∀s ∈ [x,y] (if
∃ξ ∈ [x,y] such that Df(s,x−y) = +∞ the thesis is trivial). Then ∂∞f(s,x−y) ⊂
(−∞,0), ∀s ∈ [x,y]. If, ab absurdo, f(x)−f(y) ∈ conv δ∈[x,y] {∂∞f(δ,x − y) + f(x) − f(y)}
then ∃ξ ∈ [x,y] such that f(x)−f(y) ∈ (−∞,0)+f(x)−f(y) that is 0 ∈ (−∞,0). So
f(x)−f(y) ∈ conv δ∈[x,y]∂f(δ,x−y)+conv δ∈[x,y]∂∞f(δ,x−y). Then f(x)−f(y) =
a + b where a ∈ conv δ∈[x,y]∂f(δ,x − y) and b ∈ conv δ∈[x,y]∂∞f(δ,x − y). Then
∃ξ ∈ [x,y] such that a ≤ supl∈∂f(ξ,x−y) l, that is a ≤ Df(ξ,x−y), and b ≤ 0. So the
thesis follows.
Corollary 2.2. If f is locally Lipschitz, then we have:
f(x) − f(y) ∈ conv δ∈[x,y]∂Cf(δ)(x − y).
Proof. We know that (proposition 1.4) at any point δ, ∂f(δ;x − y) = ∂C(δ)(x − y).
furthermore, form the Lipschitz hypothesis it follows easily that ∂∞f(δ;x − y) = ∅,
whenever δ. So the thesis follows.
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Approximate Jacobian matrices fro nonsmooth continuous maps and C1 optimization,

C.E.: Peano derivatives: a survey.

Nonsmooth multiobjective programming,

Optimization and Nonsmooth Analysis,

The diﬀerentiability of real functions on normed linear space using generalized subgradients.

Wets R.J-B.: The minimization of semicontinuous functions: molliﬁer subgradients.

Mean value theorem for continuous vector functions by smooth approximations

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Mean value theorem for continuous vector functions by smooth approximations

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