Vol. 29(1), 2003/2004, pp. 175–198 Adrian I. Ban and Sorin G. Gal, Department of Mathematics, University of Oradea, str. Armatei Romˆ ane 5, 3700… [616071]

Real Analysis Exchange
Vol. 29(1), 2003/2004, pp. 175–198
Adrian I. Ban and Sorin G. Gal, Department of Mathematics, University of
Oradea, str. Armatei Romˆ ane 5, 3700 Oradea, Romania.
email: [anonimizat] [anonimizat]
ON SOME POINTWISE DEFECTS OF
PROPERTIES IN REAL ANALYSIS
Abstract
In this paper we continue the research in Real Analysis, in the spirit
of the very recent book [1]. Firstly, as a refinement of the global defect
of integrability introduced in [1], §5.1, we consider here a pointwise
defect of integrability and study its properties and connections with the
pointwise defect of continuity already introduced in [1], §5.1. Secondly,
as refinements of the global defects of monotonicity and of convexity
introduced in the same book [1], §5.2, we consider and study pointwise
variants.
1 Introduction
LetUbe an abstract set and Pa given property of some elements in U.
Evidently Pdivides Uinto two disjoint sets:
UP={x∈U;xhas the property P}
and
UP={x∈U;xdoes not satisfy the property P}.
A tool of investigation of UPandUPmight be the introduction (not nec-
essarily in an unique way) of a quantity E(x)∈R, defined for all x∈U, such
that
x∈UPif and only if E(x) = 0 .
Key Words: pointwise defects of continuity, of differentiability, of integrability, of mono-
tonicity, of convexity
Mathematical Reviews subject classification: 26A15, 26A24, 26A42, 26A48, 26A51
Received by the editors November 13, 2002
175

176 Adrian I. Ban and Sorin G. Gal
In this way, for x∈UPthe quantity |E(x)|can be considered to measure
the “deviation” of xfrom the property Pand can be called the defect of
property P, atx. Of course the notion of defect is of interest only when it has
appropriate analytic properties.
In the very recent book [1] we have studied this idea in Set Theory, Topol-
ogy, Measure Theory, Real Analysis, Functional Analysis, Complex Analysis,
Algebra, Geometry, Number Theory and Fuzzy Logic.
Concerning continuity, differentiability, integrability, monotonicity and con-
vexity of real functions of one real variable, the following concepts were studied.
Definition 1.1. (i) (see, for example [1], p. 185, Definitions 5.1, 5.2) Let
f:E→Randx0∈E⊂R. The defect of continuity of fatx0is the quantity
dcont(f) (x0) = inf {δ[f(V∩E)] ;V∈ V(x0)},
where V(x0) denotes the class of all neighborhoods of x0and
δ[A] = sup {|a1−a2|;a1, a2∈A}
denotes the diameter of the set A⊂R.
Of course, dcont(f) (x0) is a very old concept in analysis, usually called mod-
ulus of oscillation of fatx0, which was introduced and studied by Bernhard
Riemann and Paul Dubois-Reymond. Here we call it defect of continuity only
for the homogeneity of language.
(ii) (see, for example [1], p. 189, Definition 5.4) Let f: [a, b]→Rand
x0∈[a, b]. The defect of differentiability of fatx0is the quantity
ddif(f) (x0) = inf {δ[F(V∩[a, b]\ {x0})] ;V∈ V(x0)},
where F: [a, b]\ {x0} →Ris defined by F(x) =f(x)−f(x0)
x−x0.
(iii) (see, for example [1], p. 190, Definition 5.6) Let f: [a, b]→Rbe
bounded. The defect of Riemann integrability of fon the interval [ a, b] is the
quantity
dint(f) ([a, b]) =/integraldisplayb
af(x)dx−/integraldisplayb
af(x)dx,
where/integraltextb
aand/integraltextb
adenote the upper and lower Darboux integrals, respectively.
We note that dint(f) ([a, b]) is only a simple rewording of a nineteenth century
criterion of Darboux.
(iv) (see, for example [1], p. 194, Definition 5.7) Let f:E→RandE⊂R.
The defect of monotonicity of fonEis the quantity
dM(f) (E) = sup {|f(x1)−f(x)|+|f(x2)−f(x)| − |f(x1)−f(x2)|;
x1, x, x 2∈E, x 1≤x≤x2}.

On Some Pointwise Defects of Properties in Real Analysis 177
(v) (see, for example [1], p. 199) Let f: [a, b]→R. The defect of convexity
offon [a, b] is the quantity
dconv(f)([a, b]) = sup {f(λx+ (1−λ)y)−(λf(x) + (1 −λ)f(y));
λ∈[0,1], x, y∈[a, b]}.
Remark 1.1. It is easily seen that while the defects in Definition 1.1, ( i) and
(ii) are pointwise ones, those in Definition 1.1, ( iii),(iv) and ( v) are global
ones.
The main aim of this paper is to refine the above global defects by defining
and studying their pointwise variants.
Section 2 deals with the pointwise defect of integrability while in Section
3 we consider pointwise defects of monotonicity. Unlike the global defect
of monotonicity in Definition 1.1, ( iv), one can use the pointwise defects to
characterize increasing and decreasing monotonicities. Section 4 deals with
pointwise defects of convexity and Section 5 contains two simple applications
to the best approximation problem. At the end some open questions are
presented in Section 6.
2 Pointwise Defect of Integrability
A pointwise version of the concept in Definition 1.1, ( iii), can be defined as
follows.
Definition 2.1. Letf: [a, b]→Rbe bounded on [ a, b]. The (pointwise)
defect of Riemann integrability of fatx0∈(a, b), is the quantity
dint(f) (x0) = lim sup
h/arrowsoutheast0/braceleftBigg
1
2h/parenleftBigg/integraldisplayx0+h
x0−hf(x)dx−/integraldisplayx0+h
x0−hf(x)dx/parenrightBigg/bracerightBigg
.
Ifx0=a, then
dint(f) (x0) = lim sup
h/arrowsoutheast0/braceleftBigg
1
h/parenleftBigg/integraldisplaya+h
af(x)dx−/integraldisplaya+h
af(x)dx/parenrightBigg/bracerightBigg
and if x0=b, then
dint(f) (x0) = lim sup
h/arrowsoutheast0/braceleftBigg
1
h/parenleftBigg/integraldisplayb
b−hf(x)dx−/integraldisplayb
b−hf(x)dx/parenrightBigg/bracerightBigg
.

178 Adrian I. Ban and Sorin G. Gal
Remark 2.1. By the definition of lim suph/arrowsoutheast0, we can write (if e.g. x0∈(a, b))
dint(f) (x0) = inf
δ>0/braceleftBigg
sup
h∈(0,δ)/braceleftBigg
1
2h/parenleftBigg/integraldisplayx0+h
x0−hf(x)dx−/integraldisplayx0+h
x0−hf(x)dx/parenrightBigg/bracerightBigg/bracerightBigg
= lim
δ/arrowsoutheast0/braceleftBigg
sup
h∈(0,δ)/braceleftBigg
1
2h/parenleftBigg/integraldisplayx0+h
x0−hf(x)dx−/integraldisplayx0+h
x0−hf(x)dx/parenrightBigg/bracerightBigg/bracerightBigg
.
The following properties hold.
Theorem 2.1. Letf, g: [a, b]→Rbe bounded on [a, b]andx0∈[a, b].
(i)0≤dint(f) (x0)≤M−m, where M= sup {f(x) ;x∈[a, b]}andm=
inf{f(x) ;x∈[a, b]}.
(ii) If fis locally Riemann integrable on x0(i.e., integrable on a subinterval
containing x0), then dint(f) (x0) = 0 . If fis Riemann integrable on
[a, b], then dint(f) (x0) = 0 , for all x0∈[a, b].
(iii) dint(f+g) (x0)≤dint(f) (x0) +dint(g) (x0).
(iv)dint(λf) (x0) =|λ|dint(f) (x0),∀λ∈R.
Proof. (i) It is immediate.
(ii) It is also immediate because the Riemann integrability implies the
equality between the lower and upper Darboux integrals.
(iii) The properties of subadditivity of upper Darboux integral, superad-
ditivity of lower Darboux integral and subadditivity of upper limit prove ( iii).
(iv) The properties
/integraldisplayd
cλf(x)dx=λ/integraldisplayd
cf(x)dx,/integraldisplayd
cλf(x)dx=λ/integraldisplayd
cf(x)dx,∀λ >0,
/integraldisplayd
cλf(x)dx=λ/integraldisplayd
cf(x)dx,/integraldisplayd
cλf(x)dx=λ/integraldisplayd
cf(x)dx,∀λ <0,
hold for every c, d∈R, c < d , and the positive homogeneity of upper limits
imply the equality.
Example 2.1. For the Dirichlet function f: [0,1]→R, defined by f(x) =
0ifxis a rational number and f(x) = 1 , otherwise, it easily follows that
dint(f) (x0) = 1 , for all x0∈[0,1].

On Some Pointwise Defects of Properties in Real Analysis 179
In what follows, consider the well-known Baire functions
M(x) = lim
δ/arrowsoutheast0Mδ(x) and m(x) = lim
δ/arrowsoutheast0mδ(x),
where
Mδ(x) = sup {f(t) ;t∈[a, b]∩(x−δ, x+δ)},
mδ(x) = inf {f(t) ;t∈[a, b]∩(x−δ, x+δ)}.
Theorem 2.2. Letf: [a, b]→Rbe bounded on [a, b]. Then, for all x0∈(a, b),
we have
dint(f) (x0) = lim
δ/arrowsoutheast0/braceleftBigg
sup
h∈(0,δ)/braceleftBigg
1
2h(L)/integraldisplayx0+h
x0−hdcont(f) (x)dx/bracerightBigg/bracerightBigg
, (1)
where (L)/integraltext
denotes the Lebesgue integral (if x0=aandx0=bin(1)appear
(L)/integraltexta+h
aand(L)/integraltextb
b−h, respectively).
Proof. By e.g. [5], p. 175–176, it follows that the Baire functions M(x), m(x)
are Lebesgue measurable and that
/integraldisplayx0+h
x0−hf(x)dx= (L)/integraldisplayx0+h
x0−hM(x)dx,
/integraldisplayx0+h
x0−hf(x)dx= (L)/integraldisplayx0+h
x0−hm(x)dx.
Consequently, by the above Remark 2.1 we get
dint(f) (x0) = lim
δ/arrowsoutheast0/braceleftBigg
sup
h∈(0,δ)/braceleftBigg
1
2h(L)/integraldisplayx0+h
x0−h(M(x)−m(x))dx/bracerightBigg/bracerightBigg
.
But by e.g. [7], p. 165 we have
dcont(f) (x0) = lim
δ/arrowsoutheast0{sup (f([a, b]∩(x0−δ, x 0+δ)))
−inf (f([a, b]∩(x0−δ, x 0+δ)))}
= lim
δ/arrowsoutheast0{Mδ(x0)−mδ(x0)}
= lim
δ/arrowsoutheast0Mδ(x0)−lim
δ/arrowsoutheast0mδ(x0)
=M(x0)−m(x0),
for all x0∈[a, b], which immediately proves the theorem.

180 Adrian I. Ban and Sorin G. Gal
Corollary 2.3. Letf: [a, b]→Rbe bounded on [a, b]. Then fis Riemann
integrable on [a, b]if and only if dint(f) (x0) = 0 , for all x0∈[a, b].
Proof. Iffis Riemann integrable on [ a, b], then by Theorem 2.1, ( ii) we get
dint(f) (x0) = 0, for all x0∈[a, b].
Now, suppose that dint(f) (x0) = 0, for all x0∈[a, b]. By (1) we get
dint(f) (x0)≥lim
δ/arrowsoutheast01
2δ(L)/integraldisplayx0+δ
x0−δdcont(f) (x)dx,
which implies lim δ/arrowsoutheast01
2δ(L)/integraltextx0+δ
x0−δdcont(f) (x)dx= 0, for all x0∈[a, b]. Since
the integrand dcont(f) (x) =M(x)−m(x) is nonnegative, we can deduce that
∀x0∈[a, b],∀ε >0,∃δx0,εsuch that for all a≤x0≤band 0 < t < δ x0,ε, if
[x0, x0+t]⊂[a, b], then
(L)/integraldisplayx0+t
x0(M(x)−m(x))dx < εt
while if [ x0−t, x0]⊂[a, b], then
(L)/integraldisplayx0
x0−t(M(x)−m(x))dx < εt.
For every ε >0, define Fεto be the collection of all intervals [ x0, x0+t]⊂[a, b]
and [ x0−t, x0]⊂[a, b] for 0 < t < δ x0,ε. Applying Cousin’s Lemma (see e.g.
[2], p. 9) there exists a partition of [ a, b],
[a, b] = [a0, a1]∪[a1, a2]∪ · · · ∪ [an−2, an−1]∪[an−1, an],
a0=a,an=b, where each [ ai, ai+1]∈Fε,i=0, n−1. Adding term by term
the inequalities for the subintervals of the partition, we obtain
(L)/integraldisplayb
a(M(x)−m(x))dx < ε (b−a),∀ε >0.
We get ( L)/integraltextb
a(M(x)−m(x))dx= 0 and because 0 ≤M(x)−m(x), it follows
M(x)−m(x) = 0, a.e. x∈[a, b]. Consequently by e.g. [5], p. 172, Theorem
1, it follows that fis almost everywhere continuous on [ a, b] and therefore it
is Riemann integrable on [ a, b].
Remark 2.2. By Theorem 2.2 and Corollary 2.3, it follows that formula (1)
can be considered in fact a generalization of the well-known result which states
that a bounded function fis Riemann integrable on [ a, b], if and only if it is
almost everywhere continuous on [ a, b]. Indeed, this immediately follows from
[1], p. 186, Theorem 5.1, ( i), which states that fis continuous on x0if and
only if dcont(f) (x0) = 0 .

On Some Pointwise Defects of Properties in Real Analysis 181
3 Pointwise Defects of Monotonicity
The pointwise variant of above Definition 1.1, ( iv), is the following.
Definition 3.1. Letf: (a, b)→Rbe a non-constant function and x0∈(a, b).
The defect of monotonicity of fonx0is the quantity
dM(f) (x0) = lim sup
εi/arrowsoutheast0,i∈{1,2}Ef,x0(ε1, ε2),
where Ef,x0(ε1, ε2) is the fraction
|f(x0−ε1)−f(x0)|+|f(x0+ε2)−f(x0)| − |f(x0−ε1)−f(x0+ε2)|
|f(x0−ε1)−f(x0+ε2)|.
Iffis constant, then by definition we take dM(f) (x0) = 0 ,∀x0∈(a, b).
Remark 3.1. Iffis a non-constant function, then obviously we can write
dM(f) (x0) = lim
δi/arrowsoutheast0/braceleftBigg
sup
εi∈(0,δi),i∈{1,2}Ef,x0(ε1, ε2)/bracerightBigg
.
The following properties can easily be proved.
Theorem 3.1. Letf: (a, b)→Randx0∈(a, b).
(i)dM(f) (x0)≥0.
(ii) If fis monotonous in a neighborhood of x0, then dM(f) (x0) = 0 .
(iii) dM(λf) (x0) =dM(f) (x0),∀λ∈R\ {0}.
(iv) If a=−b, b > 0andf(−x) =f(x),∀x∈(a, b)orf(−x) =−f(x),
∀x∈(a, b), then dM(f) (−x0) =dM(f) (x0).
(v)dM(1−f) (x0) =dM(f) (x0).
Lemma 3.2. Letf: (a, b)→Rbe locally continuous at x0∈(a, b)(i.e.,
∃ε1, ε2>0, I= (x0−ε1, x0+ε2)⊂(a, b)such that fis continuous on I). If
x0is a locally strict extremum point of f, then dM(f) (x0) = + ∞.
Proof. From the continuity of fonI= (x0−ε1, x0+ε2), there exist ε(n)
1/arrowsoutheast
0, ε(n)
2/arrowsoutheast0, such that f/parenleftBig
x0−ε(n)
1/parenrightBig
=f/parenleftBig
x0−ε(n)
2/parenrightBig
/negationslash=f(x0),∀n∈N. It
follows that for all δ1, δ2>0, sufficiently small, we have
sup
εi∈(0,δi),i∈{1,2}/braceleftbigg|f(x0−ε1)−f(x0)|+|f(x0+ε2)−f(x0)|
|f(x0−ε1)−f(x0+ε2)|

182 Adrian I. Ban and Sorin G. Gal
−|f(x0−ε1)−f(x0+ε2)|
|f(x0−ε1)−f(x0+ε2)|/bracerightbigg
= +∞,
which implies dM(f) (x0) = + ∞.
Theorem 3.3. Letf: (a, b)→Rbe continuous on (a, b), such that fis not
constant on some subintervals of (a, b). Then fis monotonic on (a, b)if and
only if dM(f) (x0) = 0 , for all x0∈(a, b).
Proof. Iffis monotonic on ( a, b), then it is immediate that dM(f) (x0) =
0,∀x0∈(a, b). Conversely, suppose that dM(f) (x0) = 0 ,∀x0∈(a, b), but f
would be not monotonic on ( a, b). Then there exist x1, x2, x3, a < x 1< x 2<
x3< b, satisfying:
(i)f(x2)< f(x1), f(x2)< f(x3)
or
(ii)f(x2)> f(x1), f(x2)> f(x3).
Case ( i). Suppose, for example, f(x1)≤f(x3) (the subcase f(x1)>
f(x3) is similar). It follows that fhas in ( x1, x3) a (locally) strict minimum
point x∗, which by Lemma 3.2 implies dM(f) (x∗) = + ∞, a contradiction.
Case ( ii). Similarly, it follows that fhas in ( x1, x3) a strict maximum
point x∗; i.e., we again get the contradiction dM(f) (x∗) = + ∞.
Remark 3.2. The condition that fcannot be constant on some subintervals
of (a, b) is necessary. Indeed, if we define f: (0,1)→Ras the continuous
polygonal line passing through the points (0 ,1),/parenleftbig1
3,1
2/parenrightbig
,/parenleftbig2
3,1
2/parenrightbig
and (1 ,1), a
simple calculation show us that dM(f) (x0) = 0 ,∀x0∈(0,1), while fis not
monotonic on (0 ,1).
Example 3.1. In [3], p. 66, the following example of nowhere monotone
function on (0 ,1) is given. Let f(x) =xifxis rational and f(x) = 1−xifx
is irrational. Let x0∈/parenleftbig
0,1
2/parenrightbig
∩Qand/parenleftBig
ε(n)
1/parenrightBig
n∈N,/parenleftBig
ε(n)
2/parenrightBig
n∈Nbe two sequences
such that x0−ε(n)
1∈(0,1)∩R\Q, x0+ε(n)
2∈(0,1)∩R\Q,∀n∈N, ε(n)
i/arrowsoutheast
0, n→ ∞ , i∈ {1,2}andε(n)
2<1−2×0,∀n∈N.We get that Ef,x0/parenleftBig
ε(n)
1, ε(n)
2/parenrightBig
is the expression
/vextendsingle/vextendsingle/vextendsingle1−x0+ε(n)
1−x0/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle1−x0−ε(n)
2−x0/vextendsingle/vextendsingle/vextendsingle−/vextendsingle/vextendsingle/vextendsingle1−x0+ε(n)
1−1 +x0+ε(n)
2/vextendsingle/vextendsingle/vextendsingle
/vextendsingle/vextendsingle/vextendsingle1−x0+ε(n)
1−1 +x0+ε(n)
2/vextendsingle/vextendsingle/vextendsingle
=1−2×0+ε(n)
1+ 1−2×0−ε(n)
2−ε(n)
1−ε(n)
2
ε(n)
1+ε(n)
2=2 (1−2×0)−2ε(n)
2
ε(n)
1+ε(n)
2.

On Some Pointwise Defects of Properties in Real Analysis 183
Passing to limit with n→ ∞ and taking into account Definition 3.1 we obtain
dM(f) (x0) = + ∞.
Letx0∈/parenleftbig1
2,1/parenrightbig
∩Qand/parenleftBig
ε(n)
1/parenrightBig
n∈N,/parenleftBig
ε(n)
2/parenrightBig
n∈Ntwo sequences as above, but
ε(n)
1<2×0−1,∀n∈N.We get
Ef,x0/parenleftBig
ε(n)
1, ε(n)
2/parenrightBig
=
/vextendsingle/vextendsingle/vextendsingle1−x0+ε(n)
1−x0/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle1−x0−ε(n)
2−x0/vextendsingle/vextendsingle/vextendsingle−/vextendsingle/vextendsingle/vextendsingle1−x0+ε(n)
1−1 +x0+ε(n)
2/vextendsingle/vextendsingle/vextendsingle
/vextendsingle/vextendsingle/vextendsingle1−x0+ε(n)
1−1 +x0+ε(n)
2/vextendsingle/vextendsingle/vextendsingle
=2×0−1−ε(n)
1+ 2×0−1 +ε(n)
2−ε(n)
1−ε(n)
2
ε(n)
1+ε(n)
2=2 (2×0−1)−2ε(n)
1
ε(n)
1+ε(n)
2.
As above we obtain dM(f) (x0) = + ∞.
Ifx0=1
2and/parenleftBig
ε(n)
1/parenrightBig
n∈N,/parenleftBig
ε(n)
2/parenrightBig
n∈Nare two sequences such that x0−ε(n)
1∈
(0,1)∩Q, x0+ε(n)
2∈(0,1)∩R\Q,∀n∈N, ε(n)
i/arrowsoutheast0, n→ ∞ , i∈ {1,2}, we get
Ef,x0/parenleftBig
ε(n)
1, ε(n)
2/parenrightBig
=ε(n)
1+ε(n)
2−/vextendsingle/vextendsingle/vextendsingleε(n)
1−ε(n)
2/vextendsingle/vextendsingle/vextendsingle
/vextendsingle/vextendsingle/vextendsingleε(n)
1−ε(n)
2/vextendsingle/vextendsingle/vextendsingle.
Therefore dM(f) (x0) = + ∞(for example, if ε(n)
1=1
n, ε(n)
2=1√
n2+1, then
limn→∞Ef,x0/parenleftBig
ε(n)
1, ε(n)
2/parenrightBig
= lim n→∞2n√
n2+1−n= +∞).
Similarly, if x0∈(0,1)∩R\Q, then dM(f) (x0) = + ∞.
More refined pointwise defects of monotonicities than those in Definition
3.1, can be introduced in such a way that can characterize the sense of mono-
tonicity. We begin with this definition.
Definition 3.2. ([7], p. 119) Let f: (a, b)→Randx0∈(a, b). We say
thatfis (pointwise) increasing at x0if∃δ >0 (sufficiently small) such that
f(x)−f(x0)
x−x0≥0,∀x/negationslash=x0,|x−x0|< δ. Analogously, fis called decreasing on
x0if∃δ >0 such thatf(x)−f(x0)
x−x0≤0,∀x/negationslash=x0,|x−x0|< δ.
Theorem 3.4. ([7], p. 120) f: (a, b)→Ris increasing (decreasing) on (a, b)
if and only if fis increasing (decreasing) at each x0∈(a, b)(in the sense of
Definition 3.2).
The pointwise deviations from the monotonicities in Definition 3.2 can be
measured by the following.

184 Adrian I. Ban and Sorin G. Gal
Definition 3.3. Letf: (a, b)→Randx0∈(a, b). The defect of increasing
monotonicity of fonx0is defined by
dIM(f) (x0) = max/braceleftbig
0,−D(f) (x0)/bracerightbig
,
where D(f) (x0) = lim supx→x0f(x)−f(x0)
x−x0.
Analogously, the defect of decreasing monotonicity of fonx0is defined by
dDM(f) (x0) = max {0, D(f) (x0)},
where D(f) (x0) = lim inf x→x0f(x)−f(x0)
x−x0.
Theorem 3.5. Letf: (a, b)→R.
(i)fcontinuous on (a, b)is increasing on (a, b)if and only if dIM(f) (x0) =
0, for all x0∈(a, b).fcontinuous on (a, b)is decreasing on (a, b)if and
only if dDM(f) (x0) = 0 , for all x0∈(a, b).
(ii) If f∈C1(a, b), then dIM(f) (x0) = max {0,−f/prime(x0)}and
dDM(f) (x0) = max {0, f/prime(x0)}, for all x0∈(a, b).
(iii) If fis increasing on (a, b), then dDM(f) (x0) =D(f) (x0), for all x0∈
(a, b). Iffis decreasing on (a, b), then dIM(f) (x0) =−D(f) (x0), for
allx0∈(a, b).
(iv) If g∈C1(a, b), f∈C1(g(a, b)), then
dIM(f◦g) (x0) =|f/prime(g(x0))|dIM(g) (x0) +g/prime(x0)dIM(f) (g(x0))
=|f/prime(g(x0))|dDM(g) (x0)−g/prime(x0)dDM(f) (g(x0))
and
dDM(f◦g) (x0) =|f/prime(g(x0))|dIM(g) (x0) +g/prime(x0)dDM(f) (g(x0))
=|f/prime(g(x0))|dDM(g) (x0)−g/prime(x0)dIM(f) (g(x0)),
for all x0∈(a, b).
(v) If f∈C1(a, b), fis invertible and f/prime(x)/negationslash= 0,∀x∈(a, b), then
dIM/parenleftbig
f−1/parenrightbig
(y0) =−dIM(f) (x0)
f/prime(x0)|f/prime(x0)|=f/prime(x0)−dDM(f) (x0)
f/prime(x0)|f/prime(x0)|
and
dDM/parenleftbig
f−1/parenrightbig
(y0) =dDM(f) (x0)
f/prime(x0)|f/prime(x0)|=f/prime(x0) +dIM(f) (x0)
f/prime(x0)|f/prime(x0)|,
for every y0=f(x0).

On Some Pointwise Defects of Properties in Real Analysis 185
(vi) If f∈C1(A, B), then for all a, b∈(A, B), a < b , there exist α, β∈(a, b)
such that
−dIM(f) (α)≤f(b)−f(a)
b−a≤dDM(f) (β).
Proof. (i) Iffis increasing on ( a, b), then by Theorem 3.4 it follows that for
every x0∈(a, b),f(x)−f(x0)
x−x0≥0, for all x/negationslash=x0, xsufficiently close to x0. This
implies D(f) (x0)≥0 and therefore dIM(f) (x0) = max/braceleftbig
0,−D(f) (x0)/bracerightbig
= 0,
for all x0∈(a, b). Conversely, by dIM(f) (x0) = 0 we get −D(f) (x0)≤0;
i.e.,D(f) (x0)≥0,∀x0∈(a, b), which by a well-known result (see e.g. [4], p.
222) implies that fis increasing on ( a, b). The proof of the second statement
is similar.
(ii) It is immediate.
(iii)f(x)−f(x0)
x−x0≥0,∀x, x 0∈(a, b), x/negationslash=x0implies D(f) (x0)≥0,∀x0∈
(a, b). Therefore dDM(f) (x0) =D(f) (x0),∀x0∈(a, b). If fis decreasing,
then the proof is similar.
(iv) Because the above property ( ii) implies dIM(h) (x0) =|h/prime(x0)|−h/prime(x0)
2
anddDM(h) (x0) =|h/prime(x0)|+h/prime(x0)
2, for every function h∈C1(a, b) and x0∈
(a, b), we have
dIM(f◦g) (x0) =|f/prime(g(x0))| |g/prime(x0)| −f/prime(g(x0))g/prime(x0)
2
=|f/prime(g(x0))|(|g/prime(x0)| −g/prime(x0)) +g/prime(x0) (|f/prime(g(x0))| −f/prime(g(x0)))
2
=|f/prime(g(x0))|dIM(g) (x0) +g/prime(x0)dIM(f) (g(x0)).
or
dIM(f◦g) (x0) =|f/prime(g(x0))| |g/prime(x0)| −f/prime(g(x0))g/prime(x0)
2
=|f/prime(g(x0))|(|g/prime(x0)|+g/prime(x0))−g/prime(x0) (|f/prime(g(x0))|+f/prime(g(x0)))
2
=|f/prime(g(x0))|dDM(g) (x0)−g/prime(x0)dDM(f) (g(x0)).
The proof of the second part is analogous.
(v) Replacing gwith f−1in the first property ( iv) we obtain
0 =dIM/parenleftbig
1(a,b)/parenrightbig
(y0) =/vextendsingle/vextendsinglef/prime/parenleftbig
f−1(y0)/parenrightbig/vextendsingle/vextendsingledIM/parenleftbig
f−1/parenrightbig
(y0)
+/parenleftbig
f−1/parenrightbig/prime(y0)dIM(f)/parenleftbig
f−1(y0)/parenrightbig

186 Adrian I. Ban and Sorin G. Gal
and
0 =dIM/parenleftbig
1(a,b)/parenrightbig
(y0) =/vextendsingle/vextendsinglef/prime/parenleftbig
f−1(y0)/parenrightbig/vextendsingle/vextendsingledDM/parenleftbig
f−1/parenrightbig
(y0)
−/parenleftbig
f−1/parenrightbig/prime(y0)dDM(f)/parenleftbig
f−1(y0)/parenrightbig
,
where 1 (a,b)(x) =x,for all x∈(a, b). Denoting y0=f(x0) the first equality
implies dIM/parenleftbig
f−1/parenrightbig
(y0) =−dIM(f) (x0)
f/prime(x0)|f/prime(x0)|and the second equality implies
dDM/parenleftbig
f−1/parenrightbig
(y0) =dDM(f) (x0)
f/prime(x0)|f/prime(x0)|.The proof of the others equalities is similar
starting from the property dDM/parenleftbig
1(a,b)/parenrightbig
(y0) = 1 ,∀y0∈(a, b).
(vi) Because|f/prime(x)|+f/prime(x)
2≥f/prime(x),∀x∈(a, b), we get/integraltextb
adDM(f) (x)dx
≥f(b)−f(a).On the other hand, there exists β∈(a, b) such that/integraltextb
adDM(f) (x)dx= (b−a)dDM(f) (β). These imply the desired inequality.
The proof of the other inequality is similar.
Example 3.2. Iffis not continuous on ( a, b), then Theorem 3.5, ( i), fails to
be valid. Indeed, let f: (0,1)→Rbe defined by f(x) = 0 if xis rational and
f(x) = 1 if xis irrational. If x0∈Q∩(0,1), then
sup/braceleftbiggf(x)−f(x0)
x−x0;x∈(x0−δ, x 0+δ)∩(0,1), x/negationslash=x0/bracerightbigg
≥sup/braceleftbiggf(x)
x−x0;x∈(R\Q)∩(x0−δ, x 0+δ)∩(0,1)/bracerightbigg
= +∞,∀δ >0.
Therefore
D(f) (x0) = lim sup
x→x0f(x)−f(x0)
x−x0= +∞.
We get dIM(f) (x0) = 0 .
Ifx0∈(R\Q)∩(0,1),then
sup/braceleftbiggf(x)−f(x0)
x−x0;x∈(x0−δ, x 0+δ)∩(0,1), x/negationslash=x0/bracerightbigg
≥sup/braceleftbiggf(x)−1
x−x0;x∈Q∩(x0−δ, x 0+δ)∩(0,1)/bracerightbigg
= +∞,∀δ >0.
Therefore
D(f) (x0) = lim sup
x→x0f(x)−f(x0)
x−x0= +∞.
We get dIM(f) (x0) = 0 .
As above we obtain D(f) (x0) =−∞ anddDM(f) (x0) = 0 .

On Some Pointwise Defects of Properties in Real Analysis 187
Remark 3.3. The properties in Theorem 3.5, ( iv),(v), can be considered
generalizations of the well-known results which state that the composition of
two increasing (decreasing) functions is also increasing, the composition of an
increasing function with a decreasing function is a decreasing function, the
inverse of an increasing function is increasing and the inverse of a decreasing
function is decreasing.
4 Pointwise Defect of Convexity
A pointwise analogue of the global defect of convexity in Definition 1.1, ( v),
might be the following.
Definition 4.1. Letf: (a, b)→Randx0∈(a, b). The pointwise defect of
convexity of fatx0is the quantity
dconv(f) (x0) =
lim sup
x1,x2→x0/braceleftBigg
sup
λ∈[0,1]f(λx1+ (1−λ)x2)−λf(x1)−(1−λ)f(x2)
(x1−x2)2/bracerightBigg
.
Analogously, the pointwise defect of concavity of fonx0is the quantity
dconc(f) (x0) =
lim sup
x1,x2→x0/braceleftBigg
sup
λ∈[0,1]λf(x1) + (1 −λ)f(x2)−f(λx1+ (1−λ)x2)
(x1−x2)2/bracerightBigg
.
We present properties of these defects.
Theorem 4.1. Letf: (a, b)→Randx0∈(a, b).
(i)dconv(f) (x0)≥0anddconc(f) (x0)≥0.
(ii) If fis convex on (a, b), then dconv(f) (x0) = 0 . Iffis concave on (a, b),
then dconc(f) (x0) = 0 .
(iii) If fis strongly concave on (a, b); i.e., there exists M > 0such that
Mλ(1−λ) (x1−x2)2≤f(λx1+ (1−λ)x2)−λf(x1)−(1−λ)f(x2),
(2)
for all λ∈[0,1], x1, x2∈[a, b], then dconv(f) (x0)≥M
4, where
M= sup {M;Mverifies (2)}.
(iv) If fis locally convex on x0(i.e., convex in a neighborhood of x0), then
dconv(f) (x0) = 0 .

188 Adrian I. Ban and Sorin G. Gal
(v) If [x1, λx 1+ (1−λ)x2, x2;f]denotes the divided difference, then
dconv(f) (x0) = lim sup
x1,x2→x0/braceleftBigg
sup
λ∈[0,1]−λ(1−λ) [x1, λx 1+ (1−λ)x2, x2;f]/bracerightBigg
dconc(f) (x0) = lim sup
x1,x2→x0/braceleftBigg
sup
λ∈[0,1]λ(1−λ) [x1, λx 1+ (1−λ)x2, x2;f]/bracerightBigg
.
(vi)dconv(Ax+B) (x0) =dconc(Ax+B) (x0) = 0 ,∀A, B∈R,∀x0∈R.
(vii) dconv(−f) (x0) =dconc(f) (x0).
(viii) If a=−b, b > 0andf(−x) =f(x),∀x∈(a, b), then dconv(f) (−x0)
=dconv(f) (x0)anddconc(f) (−x0) =dconc(f) (x0).
(ix) If a=−b, b > 0andf(−x) =−f(x),∀x∈(a, b), then dconv(f) (−x0)
=dconc(f) (x0)anddconc(f) (−x0) =dconv(f) (x0).
Proof. (i) They are immediate because for x1/negationslash=x2andλ= 0 or λ= 1, we
getf(λx1+ (1−λ)x2)−λf(x1)−(1−λ)f(x2) = 0 .
(ii)f(λx1+ (1−λ)x2)−λf(x1)−(1−λ)f(x2)≤0,∀λ∈[0,1],∀x1, x2∈
[a, b] immediately implies dconv(f) (x0) = 0 ,∀x0∈(a, b). Similarly if fis
concave.
(iii),(iv) Are immediate.
(v) Simple calculations show that for x1/negationslash=x2,
λf(x1) + (1 −λ)f(x2)−f(λx1+ (1−λ)x2)
(x1−x2)2=
λ(1−λ) [x1, λx 1+ (1−λ)x2, x2;f].
(vi),(vii) Are obvious.
(viii)
dconv(f) (−x0) =
lim sup
x1,x2→−x0/braceleftBigg
sup
λ∈[0,1]f(λx1+ (1−λ)x2)−λf(x1)−(1−λ)f(x2)
(x1−x2)2/bracerightBigg
= lim sup
y1,y2→x0/braceleftBigg
sup
λ∈[0,1]f(−λy1−(1−λ)y2)−λf(−y1)−(1−λ)f(−y2)
(−y1+y2)2/bracerightBigg
= lim sup
y1,y2→x0/braceleftBigg
sup
λ∈[0,1]f(λy1+ (1−λ)y2)−λf(y1)−(1−λ)f(y2)
(−y1+y2)2/bracerightBigg
=dconv(f) (x0).

On Some Pointwise Defects of Properties in Real Analysis 189
The proof of the second equality is similar.
(ix) It is similar to ( viii).
Example 4.1. The function f(x) =−x2is strongly concave on (a, b), a, b∈
R, with M∈(0,1]. We obtain dconv(f) (x0) =1
4,∀x0∈(a, b)that is the
equality in Theorem 4.1, property (iii).
Example 4.2. Forf: [−1,1]→R, f(x) =|x|, we easily get
dconv(f) (x0) = 0 ,∀x0∈(−1,1), dconc(f) (x0) = 0 ,∀x0∈(−1,0)∪(0,1)and
dconc(f) (0) = + ∞.
Corollary 4.2. Iff∈C2(a, b), then for all x0∈(a, b)we have
dconv(f) (x0) = max/braceleftbigg
0,−f/prime/prime(x0)
8/bracerightbigg
,
dconc(f) (x0) = max/braceleftbigg
0,f/prime/prime(x0)
8/bracerightbigg
.
Proof. We have
sup
λ∈[0,1]{−λ(1−λ) [x1, λx 1+ (1−λ)x2, x2;f]}
= max/braceleftBigg
0,sup
λ∈(0,1)λ(1−λ) (−[x1, λx 1+ (1−λ)x2, x2;f])/bracerightBigg
.
Without loss of generality, we can suppose x1< x 2. By the mean value theo-
rem, there exists ξλ∈(x1, x2) with −[x1, λx 1+ (1−λ)x2, x2;f] =−f/prime/prime(ξλ)
2,
∀λ∈(0,1), which implies
dconv(f) (x0) = max/braceleftBigg
0,lim sup
x1,x2→x0/braceleftBigg
sup
λ∈(0,1)λ(1−λ)
2(−f/prime/prime(ξλ))/bracerightBigg/bracerightBigg
.
But
inf
x∈(x1,x2)(−f/prime/prime(x))≤ −f/prime/prime(ξλ)≤ sup
x∈(x1,x2)(−f/prime/prime(x)), (3)
which implies
inf
x∈(x1,x2)(−f/prime/prime(x)) sup
λ∈(0,1)λ(1−λ)
2≤sup
λ∈(0,1)/braceleftbiggλ(1−λ)
2(−f/prime/prime(ξλ))/bracerightbigg
.
From supλ∈(0,1)λ(1−λ)
2=1
8, by passing above to lim supx1,x2→x0and taking
into account the continuity of f/prime/primeon (a, b), we get
−f/prime/prime(x0)
8≤lim sup
x1,x2→x0/braceleftBigg
sup
λ∈(0,1)λ(1−λ)
2(−f/prime/prime(ξλ))/bracerightBigg
. (4)

190 Adrian I. Ban and Sorin G. Gal
Concerning f/prime/prime(x0) we have three possibilities: ( i)f/prime/prime(x0)<0; (ii)f/prime/prime(x0)>
0; (iii)f/prime/prime(x0) = 0 .
Case ( i). There exists a neighborhood V0ofx0such that f/prime/prime(x)<0,∀x∈
V0, which implies supx∈(x1,x2)(−f/prime/prime(x))>0, for all x1, x2∈V0. By (3) we
obtain
sup
λ∈(0,1)/braceleftbiggλ(1−λ)
2(−f/prime/prime(ξλ))/bracerightbigg
≤ sup
x∈(x1,x2)(−f/prime/prime(x)) sup
λ∈(0,1)/braceleftbiggλ(1−λ)
2/bracerightbigg
,
for all x1, x2∈V0, x1< x 2, wherefrom passing to lim supx1,x2→x0, we get
lim sup
x1,x2→x0/braceleftBigg
sup
λ∈(0,1)λ(1−λ)
2(−f/prime/prime(ξλ))/bracerightBigg
≤ −f/prime/prime(x0)
8.
Combined with (4) it follows that
dconv(f) (x0) =−f/prime/prime(x0)
8= max/braceleftbigg
0,−f/prime/prime(x0)
8/bracerightbigg
.
Case ( ii). There exists a neighborhood V0ofx0such that f/prime/prime(x)>0,∀x∈
V0, which by (3) implies −f/prime/prime(ξλ)<0, for all x1, x2∈V0, and therefore
lim sup
x1,x2→x0/braceleftBigg
sup
λ∈(0,1)λ(1−λ)
2(−f/prime/prime(ξλ))/bracerightBigg
≤0.
As a consequence,
dconv(f) (x0) = 0 = max/braceleftbigg
0,−f/prime/prime(x0)
8/bracerightbigg
.
Case ( iii). By hypothesis we have f/prime/prime(x0) = 0. By (4) it follows
0≤lim sup
x1,x2→x0/braceleftBigg
sup
λ∈(0,1)λ(1−λ)
2(−f/prime/prime(ξλ))/bracerightBigg
.
Suppose that
0< l= lim sup
x1,x2→x0/braceleftBigg
sup
λ∈(0,1)λ(1−λ)
2(−f/prime/prime(ξλ))/bracerightBigg
.
Then, for 0 < l1< l, there exists a neighborhood V0ofx0such that we have
0< l1<sup
λ∈(0,1){λ(1−λ) (−[x1, λx 1+ (1−λ)x2, x2;f])},

On Some Pointwise Defects of Properties in Real Analysis 191
for all x1, x2∈V0. It follows that there exists λ0∈(0,1) (depending on x1, x2
too) such that
0< l1< λ 0(1−λ0) (−[x1, λ0x1+ (1−λ0)x2, x2;f]) ;
i.e.,
0<l1
λ0(1−λ0)<−[x1, λ0x1+ (1−λ0)x2, x2;f],∀x1, x2∈V0.
Butl1
λ0(1−λ0)≥4l1>0. Passing here to limit with x1, x2→x0, it follows
0<4l1≤l1
λ0(1−λ0)≤ −f/prime/prime(x0)
2;
that is, f/prime/prime(x0)<0, a contradiction. As a conclusion,
lim sup
x1,x2→x0/braceleftBigg
sup
λ∈(0,1)λ(1−λ)
2(−f/prime/prime(ξλ))/bracerightBigg
= 0,
which implies
dconv(f) (x0) = max {0,0}= 0 = max/braceleftbigg
0,−f/prime/prime(x0)
8/bracerightbigg
.
The proof of the second formula in statement is similar, which proves the
theorem.
As an immediate consequence we obtain the next assertion.
Corollary 4.3. Letf∈C2(a, b).
(i)fis convex on (a, b)if and only if dconv(f) (x0) = 0 , for all x0∈(a, b).
(ii)fis concave on (a, b)if and only if dconc(f) (x0) = 0 , for all x0∈(a, b).
(iii) dIM(f/prime) (x0) = 8 dconv(f) (x0)anddDM(f/prime) (x0) = 8 dconc(f) (x0), for
allx0∈(a, b).
The results proved in Corollary 4.2 are also used in the proof of following
result.

192 Adrian I. Ban and Sorin G. Gal
Corollary 4.4. (i)Ifg∈C2(a, b), f∈C2(g(a, b)), then
dconv(f◦g) (x0)≤(g/prime(x0))2dconv(f) (g(x0)) +|f/prime(g(x0))|dconv(g) (x0)
+1
8g/prime/prime(x0)dIM(f) (g(x0))
and
dconc(f◦g) (x0)≤(g/prime(x0))2dconc(f) (g(x0)) +|f/prime(g(x0))|dconc(g) (x0)
−1
8g/prime/prime(x0)dIM(f) (g(x0)),
for every x0∈(a, b).
(ii)Iff∈C2(a, b), fis invertible and f/prime(x)/negationslash= 0,∀x∈(a, b), then
dconv/parenleftbig
f−1/parenrightbig
(y0)≥f/prime/prime(x0)dIM(f) (x0)−8dconv(f) (x0)
8|f/prime(x0)|3
and
dconc/parenleftbig
f−1/parenrightbig
(y0)≥ −f/prime/prime(x0)dIM(f) (x0) + 8dconc(f) (x0)
8|f/prime(x0)|3,
for every y0=f(x0), x0∈(a, b).
Proof. (i) Because the property in Corollary 4.2 implies dconv(h) (x)
=|h/prime/prime(x)|−h/prime/prime(x)
16,∀x∈(a, b), for every function h∈C2(a, b), we get
dconv(f◦g) (x0) =/vextendsingle/vextendsingle/vextendsinglef/prime/prime(g(x0)) (g/prime(x0))2+f/prime(g(x0))g/prime/prime(x0)/vextendsingle/vextendsingle/vextendsingle
16
−/parenleftBig
f/prime/prime(g(x0)) (g/prime(x0))2+f/prime(g(x0))g/prime/prime(x0)/parenrightBig
16
≤|f/prime/prime(g(x0))|(g/prime(x0))2+|f/prime(g(x0))| |g/prime/prime(x0)|
16
−f/prime/prime(g(x0)) (g/prime(x0))2−f/prime(g(x0))g/prime/prime(x0)
16
= (g/prime(x0))2|f/prime/prime(g(x0))| −f/prime/prime(g(x0))
16+|f/prime(g(x0))||g/prime/prime(x0)| −g/prime/prime(x0)
16
+1
8g/prime/prime(x0)|f/prime(g(x0))| −f/prime(g(x0))
2

On Some Pointwise Defects of Properties in Real Analysis 193
= (g/prime(x0))2dconv(f) (g(x0)) +|f/prime(g(x0))|dconv(g) (x0)
+1
8g/prime/prime(x0)dIM(f) (g(x0)),∀x0∈(a, b).
The proof of the second inequality is similar.
(ii) Because dconv/parenleftbig
1(a,b)/parenrightbig
(x) = 0 ,∀x∈(a, b), where 1 (a,b)is the identical
function on ( a, b), taking g=f−1in (i) we obtain
/parenleftBig/parenleftbig
f−1(y0)/parenrightbig/prime/parenrightBig2
dconv(f)/parenleftbig
f−1(y0)/parenrightbig
+/vextendsingle/vextendsinglef/prime/parenleftbig
f−1(y0)/parenrightbig/vextendsingle/vextendsingledconv/parenleftbig
f−1/parenrightbig
(y0)
+1
8/parenleftbig
f−1/parenrightbig/prime/prime(y0)dIM(f)/parenleftbig
f−1(y0)/parenrightbig
≥0,∀y0∈f((a, b));
that is,
dconv/parenleftbig
f−1/parenrightbig
(y0)≥f/prime/prime(x0)dIM(f) (x0)−8dconv(f) (x0)
8|f/prime(x0)|3
for every y0=f(x0), x0∈(a, b).The proof of the second inequality is similar.
Remark 4.1. For example, the first formula in Corollary 4.3 ( iii) above can be
viewed as a generalization of the following well-known result in Real Analysis.
fis convex on ( a, b) if and only if f/primeis increasing on ( a, b).
The above considerations and the concept of convex (concave) function of
order n∈ {− 1,0,1,2, . . .}on (a, b) in [6], allow us to introduce the following.
Definition 4.2. Letf: (a, b)→Randx0∈(a, b). The pointwise defect of
convexity of order noffatx0is the quantity
d(n)
conv(f) (x0) = max

0,− lim sup
xi→x0,i∈{1,…,n +2}
xi/negationslash=xj,i/negationslash=j[x1, . . . , x n+2;f]

.
Analogously, the pointwise defect of concavity of order noffatx0is the
quantity
d(n)
conc(f) (x0) = max

0, lim sup
xi→x0,i∈{1,…,n +2}
xi/negationslash=xj,i/negationslash=j[x1, . . . , x n+2;f]

.
Here [ x1, . . . , x n+2;f] denotes the divided difference.

194 Adrian I. Ban and Sorin G. Gal
Remark 4.2. Iff∈Cn+1(a, b), then by the mean value for [ x1, . . . , x n+2;f]
in [6], we immediately get
d(n)
conv(f) (x0) = max/braceleftbigg
0,−f(n+1)(x0)
(n+ 1)!/bracerightbigg
,
d(n)
conc(f) (x0) = max/braceleftbigg
0,f(n+1)(x0)
(n+ 1)!/bracerightbigg
,
for every x0∈(a, b), such that for n= 0 and n= 1 we essentially recapture
the pointwise defects in Definitions 3.3 and 4.1.
5 Applications
In what follows we present some simple applications. Let
IM(x0) ={g: (a, b)→R;gis differentiable and increasing on x0},
where x0∈(a, b) and the pointwise increasing monotonicity is defined as in
Definition 3.2, let
IM(a, b) ={g: (a, b)→R;gis differentiable and increasing on ( a, b)},
forfdifferentiable on x0∈(a, b), let
EIM(f) (x0) = inf {dIM(f−g) (x0) ;g∈IM(x0)},
forfdifferentiable on ( a, b), let
/bardblf/bardblIM= sup {dIM(f) (x) ;x∈(a, b)}and let
EIM(f) (a, b) = inf {/bardblf−g/bardblIM;gis differentiable and increasing on ( a, b)}.
Remark 5.1. /bardbl·/bardblIMis a special kind of norm, because /bardblf/bardblIM= 0 if and
only if fis monotonically increasing on ( a, b),/bardblλf/bardblIM=λ/bardblf/bardblIMonly for
λ≥0,/bardblf+g/bardblIM≤ /bardblf/bardblIM+/bardblg/bardblIM.
Theorem 5.1. Letf: (a, b)→R.
(i) If fis differentiable on x0∈(a, b), then EIM(f) (x0)≥dIM(f) (x0).
(ii) If fis differentiable on (a, b), then EIM(f) (a, b)≥ /bardblf/bardblIM.

On Some Pointwise Defects of Properties in Real Analysis 195
Proof. (i) For any g∈IM(x) we have
dIM(f) (x0) =|f/prime(x0)| −f/prime(x0)
2
=|f/prime(x0)| − |g/prime(x0)|
2+|g/prime(x0)| −g/prime(x0)
2+g/prime(x0)−f/prime(x0)
2
=|f/prime(x0)| − |g/prime(x0)|
2+g/prime(x0)−f/prime(x0)
2
≤|f/prime(x0)−g/prime(x0)|
2−f/prime(x0)−g/prime(x0)
2=dIM(f−g) (x0).
Passing to infimum with g∈IM(x0) we get ( i).
(ii) Passing to supremum with x∈(a, b) in ( i) we get
/bardblf/bardblIM≤sup
x∈(a,b){inf{dIM(f−g) (x) ;g∈IM(x)}}
≤ inf
g∈IM(a,b){sup{dIM(f−g) (x) ;x∈(a, b)}}=EIM(f) (a, b),
which proves the theorem.
Let
CONV (x0) =/braceleftbig
g: (a, b)→R;g∈C2(a, b) and gis convex on x0/bracerightbig
,
where x0∈Xand the pointwise convexity in x0is as in Theorem 4.1, ( iv),
forg∈C2(a, b),
ECONV (f) (x0) = inf {dconv(f−g) (x0) ;g∈CONV (x0)},
/bardblf/bardblCONV = sup {dconv(f) (x) ;x∈(a, b)},
(/bardbl·/bardblCONVis a special kind of norm, because /bardblf/bardblCONV= 0 if and only if f
is convex on ( a, b),/bardblλf/bardblCONV=λ/bardblf/bardblCONVonly for λ≥0,/bardblf+g/bardblCONV≤
/bardblf/bardblCONV+/bardblg/bardblCONV),
ECONV (f) (a, b) = inf/braceleftbig
/bardblf−g/bardblCONV;g∈C2(a, b), gis convex on ( a, b)/bracerightbig
.
As was done above, we can prove the following.
Theorem 5.2. Letf∈C2(a, b). We have:
(i)ECONV (f) (x)≥dconv(f) (x),∀x∈(a, b).
(ii)ECONV (f) (a, b)≥ /bardblf/bardblCONV.

196 Adrian I. Ban and Sorin G. Gal
Proof. From Corollary 4.2, we have
dconv(f) (x0) = max/braceleftbigg
0,−f/prime/prime(x0)
8/bracerightbigg
=|f/prime/prime(x0)| −f/prime/prime(x0)
16.
Reasoning as in the proof of the theorem above, we get the desired conclusion.
6 Open Problems
Concerning the above results, the study of the following questions would be
of interest.
Question 1. What connections exist between dIM(f/prime) (x0) and dconv(f) (x0)
in Corollary 4.3, ( iii) when fis only in C1(a, b) or only differentiable on ( a, b)
(and it is not in C2(a, b))? We conjecture something of the form
M1dIM(f/prime) (x0)≤dconv(f) (x0)≤M2dIM(f/prime) (x0),
where M1, M2are independent of x0∈(a, b).
Question 2. Do Theorems 5.1 and 5.2 remain valid in the case when the func-
tions fandgin the definitions of EIM(f) (x0), EIM(f) (a, b), ECONV (f) (x0),
ECONV (f) (a, b) are supposed to be non- smooth; i.e., are only continuous?
Question 3. Are Theorem 3.5, ( vi) valid in the case when fis only continu-
ous, and Corollary 4.4, in the case when fandgare only of C1-class or only
differentiable?
References
[1] A. I. Ban and S. G. Gal, Defects of Properties in Mathematics. Quantitative
Characterizations , World Scientific, New Jersey, London, Singapore, Hong
Kong, 2002.
[2] A. M. Bruckner, J. B. Bruckner and B. S. Thomson, Real Analysis , Inter-
national Edition Upper Saddle River, NJ, Prentice Hall, 1997.
[3] B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis (in
Romanian), Editura S ¸tiint ¸ific˘ a, Bucharest, 1973.
[4] V. Hiri¸ s, M. Megan and C. Popa, Introduction to Mathematical Analysis by
Exercises and Problems (in Romanian), Editura Facla, Timi¸ soara, 1976.

On Some Pointwise Defects of Properties in Real Analysis 197
[5] I. P. Natanson, The Theory of Real Variable Functions (in Romanian),
Editura Tehnic˘ a, Bucharest, 1957.
[6] T. Popoviciu, Sur quelques propri´ et´ es des functions d’une o` u de deux vari-
ables r´ eeles , Mathematica (Cluj), 8(1933), 1–85.
[7] G. Siret ¸chi, Differential and Integral Calculus (in Romanian), vol. I, Edi-
tura S ¸tiint ¸ific˘ a ¸ si Enciclopedic˘ a, Bucharest, 1985.

198