Comp. Appl. Math. (2013) 32:435–446
DOI 10.1007/s40314-013-0049-z
Some continuity notions for interval functions
and representation
Benjamín Bedregal · Regivan Santiago
Received: 13 March 2012 / Revised: 6 May 2013 / Accepted: 17 May 2013 / Published online: 15 June 2013
© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract The formalization of correctness and optimality concepts, which are desirable
requirements for interval (and other self-validated methods) libraries, is related to topological
aspects of real numbers and Moore arithmetics, characterizing some families of interval
functions. The main motivation is that real continuous functions are largely used in many
fields of human activity, and a characterization of their interval representation in terms of
interval topologies leads to the knowledge of how interval algorithms are suitable to represent
those functions. In this paper, we characterize and relate some classes of interval functions
with respect to three natures of an interval (as a set, as an information, as a number). These
natures establish different ways to classify intervals, and hence different notions of continuity.
Here we relate the notion of interval representations with those classes of functions.
Keywords Interval analysis · Interval representation · Semantics of intervals ·
Bi-continuity · Correctness · Optimality
Mathematics Subject Classification (2000)
Primary 65G40; Secondary 54C05
Communicated by Renata Hax Reiser Sander.
The partial results described in this paper were previously presented in IntMath TDS at CNMAC
2010-SBMAC honoring Prof. Dalcidio Claudio and his leadership over more than 40 years of Interval
Mathematics research in Brazil.
B. Bedregal · R. Santiago (B)
Group for Logic, Language, Information, Theory and Application–LoLITA,
Departamento de Informática e Matemática Aplicada–DIMAp,
Universidade Federal do Rio Grande do Norte–UFRN,
Campus Universitário, Natal CEP 59072-970, Brazil
e-mail: regivan@dimap.ufrn.br
B. Bedregal
e-mail: bedregal@dimap.ufrn.br
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B. Bedregal, R. Santiago
1 Introduction
An interval is a continuum of real numbers defined by its endpoints. Interval analysis, proposed by Moore in the 50s, concerns the discovery of interval functions to produce bounds
on the accuracy of numerical results that are guaranteed to be sharp and correct. The last
criterion, correctness, is the main one since it establishes that the result of an interval computation must always contain the value of the related real function. Although correctness is
a desirable property, not every interval method is correct. Santiago et al. (2006) formalizes
the notion of correctness for interval functions and shows that the class of these functions is
related with another class that is broader than that of real continuous functions.
The notion of continuity plays a fundamental role in many fields of human activity. Moore
(1962) introduced a metric distance for the set of real intervals and therefore a continuity
notion for interval functions. He also showed that each real continuous function can be
extended to a continuous interval function. On the other hand, Scott (1970) showed that IR
endowed with the reverse of inclusion order is a continuous (but not algebraic) domain. This
structure has a continuity notion which is the same from the topological and order viewpoint
(Acióly 1991; Smyth 1992; Stoltenberg-Hansen 1994). Later, Acióly and Bedregal (1997)
defined a quasi-metric for IR and proved that the induced topology is, exactly, the Scott
topology for IR and the subjacent metric space is that of Moore. Santiago et al. (2005)
showed some relations between continuous functions of those topologies, since they are the
most accepted topologies for intervals.
In this paper, we resume this research by considering a new notion of continuity for intervals which captures the essence of an interval as a set. This notion together with those explored
by Santiago et al. (2006) is a consequence of three semantical viewpoints of an interval: an
interval as a generalization of a number, as an information and as a set. These viewpoints
induce, respectively, Moore topology, Scott topology, and the topology induced by the dual
Scott topology. The set and informational viewpoints induce the notion of bi-continuity which
is also explored here. The notion of interval representation is then related to those different
notions of continuous interval functions. This enables us to understand the role of an interval
function when it is, for example, described in any specification language.
Section 2 shows the connection between the three interpretations of intervals and the
topological counterparts. Section 3 characterizes the families of continuous functions which
come from these three viewpoints of intervals. Section 4 shows the connection between
correctness and optimality, formalized in Santiago et al. (2006) with the concept of canonical
interval representation, and the viewpoints of intervals. For more detailed information about
topology and domain theory, the reader is referred to Dugundji (1966), Sieradski (1992),
Smyth (1992), Gierz (2003), and Stoltenberg-Hansen (1994), respectively.
2 Distances and continuities for intervals
In this section, we present a brief overview of distances, and hence of continuous functions,
that are derived from each interpretation of intervals.
The concept of distance in a set A is formalized by the notion of quasi-metric. A metric
is a function d : A × A → R, such that for all a, b, c ∈ A, d(a, b) = 0 ⇔ a = b,
d(a, b) = d(b, a), and d(a, c) ≤ d(a, b) + d(b, c). The pair (A, d) is called metric space.
On the set of real numbers, the notion of distance between two real numbers is given by the
function dr (r, s) =| r − s |.
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A quasi-metric generalizes the notion of metric, in the sense that it is a function d :
A × A → R, such that (a) d(a, a) = 0, (b) d(a, c) ≤ d(a, b) + d(b, c) and (c) d(a, b) =
d(b, a) = 0 ⇒ a = b.1 A quasi-metric space is a pair (A, q), where A is a set and q a
quasi-metric over A. For every quasi-metric q, it is always possible to define another quasimetric called conjugated quasi-metric defined by q(a, b) = q(b, a) and a metric q ∗ , such
that
q ∗ (a, b) = max{q(a, b), q(a, b)}.
(1)
Since quasi-metrics generalize the notion of distance, it is also possible to define, from a quasimetric q, an open ball B(a, ) = {s ∈ A : q(s, a)<},2 B(a, ) = {s∈A : q(s, a) < } and
B ∗ (a, ) = {s∈A : q ∗ (s, a) < }. These three kinds of balls define three topological spaces
induced from quasi-metrics T (q) and T (q), and the metric T (q ∗ ).3
Lemma 2.1 A subset O of a (quasi-)metric space A is an open set if and only if for all
x ∈ O, there exists > 0, such that B(x, ) ⊆ O.
Proof Straightforward.
Every distance d : A × A → R determines the following notion of continuity: f : A −→
A is d-continuous, if for each x ∈ A and > 0 there exists δ > 0 such that for all y ∈ A,
d(x, y) ≤ δ ⇒ d( f (x), f (y)) ≤ (2)
Proposition 2.2 Let d be a distance (quasi-metric), then f : A −→ A is continuous w.r.t.
(2) iff it is continuous w.r.t. the topology induced by d.
A closed interval of real numbers or just an interval can be interpreted as:
1. an extension of a real number,
2. an information about a real number, or
3. a set of real numbers.
There are three natural notions of distances for intervals, each comes from the interpretation of what is an interval. In the first viewpoint, the set of intervals, IR = {[a, b] : a ≤ b},
is seen as a subset of R × R; therefore, IR is a sub-metric space of R × R and so it admits
several equivalent distances.
Moore (1962) chose for IR the following distance:
dM ([a, b], [c, d]) = max{| a − c |, | b − d |}
(3)
Continuous functions with respect to dM will be called Moore-continuous.
In the second viewpoint, the distance cannot be symmetric, since the distance of a better
information X about a real number w.r.t. a worst Y , i.e., X ⊆ Y , should be 0, because it is
always possible to infer, without effort, the worst from the better, but the converse is not so
1 Observe that d(a, b) = d(b, a) = 0 and a = b do not hold, but d(a, b) = d(b, a) = 0 and a = b, and
d(a, b) = d(b, a) = 0 and a = b can hold.
2 Note that we use q(s, a) < instead of the usual q(a, s) < [see Smyth (1992, p.700)]. This is done to
make the resulting open balls compatible with open sets of Interval Scott Topology which capture the intuition
stated in axiom (4) below.
3 Note that when q is a metric, q, q, and q ∗ coincide and the topological spaces are the same.
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B. Bedregal, R. Santiago
easy, since it requires more information (Smyth 1992; Acióly and Bedregal 1997). Therefore,
this kind of distance should satisfy the following axiom:
d(X, Y ) = 0 if and only if X ⊆ Y
(4)
The distance for intervals satisfying (4) was introduced by Acióly and Bedregal (1997), where
they defined the following function:
dI ([a, b], [c, d]) = max{c − a, b − d, 0}
(5)
They proved that dI is a quasi-metric and that the continuous functions with respect to dI are
exactly the Scott-continuous functions, namely F is Scott/dI -continuous if for all directed
set ⊆ IR,
F
=
F()
In fact, the topology associated to dI is exactly the Scott topology on the continuous domain
IR endowed with the information order which is the opposite order of inclusion of sets (Scott
1970).
The third viewpoint, is the dual of the information viewpoint, namely an interval is an
extension (a set) of real numbers; in other words, the natural order relation associated with
this viewpoint is the inclusion of sets. Hence, the derived natural definition for distance is
the conjugated quasi-metric for dI :
dE ([a, b], [c, d]) = dI ([a, b], [c, d]) = dI ([c, d], [a, b]) = max{a − c, d − b, 0}
(6)
In other words, this viewpoint considers intervals as elements of the opposite partial order
IRop which is endowed with the conjugated quasi-metric dI . The resulting topological space
is exactly IRop endowed with the respective Scott topology. The connection between the
metric viewpoint and the duality between extension and information viewpoints is captured
in the following proposition (c.f. Acióly and Bedregal 1997):
Proposition 2.3 dI∗ = dE∗ = dM .
Observe that these notions of continuity are complementary, whereas the continuity based
on dI preserves the supremum of nested chain of intervals, dE preserves the infimum. Therefore, there is a close relation between those viewpoints of intervals and the subjacent notion
of distance. Acióly and Bedregal (1997) as well as Santiago et al. (2005, 2006) compared
those aspects of continuity and showed an important result that the intersection of those
classes of functions is not empty but one is not a superset of the other (c.f. Fig. 1). In the next
section, we show a subclass of this intersection which perfectly inherits aspects of dI , dE and
dM . The classes os dI , dE and dM continuous function will be denoted by DI , DE and DM ,
respectively.
3 Characterizing dI , dE and dM -continuities
In this section, we give a classification of interval functions with respect to the notions of
continuities described above. We also characterize a class of functions which is a subset of
the intersection between interval functions which are Moore- and Scott-continuous, namely
the class of bi-continuous interval functions. Those functions are very interesting, since
they inherit the three aspects of interval continuity described in Sect. 2, namely metric,
information, and extension.
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Definition 3.1 A function F : IR −→ IR is bi-continuous if it is dI and dE -continuous.
Using domain theory language, bi-continuous functions preserve the supremum and infimum of bi-directed sets.4
The next results relate four classes of continuous interval functions: Proposition 3.2 states
that there are Scott- continuous functions which neither do preserve infima nor are also
Moore-continuous. Proposition 3.4 states that that there are dE -continuous functions which
neither do preserve suprema nor are also Moore-continuous. Proposition 3.6 states that
there exist Moore-continuous functions which neither do preserve infima nor do preserve
suprema. Proposition 3.9 states that every bi-continuous function is also Moore-continuous.
And Proposition 3.6 states that not every Moore-continuous function is Scott-continuous and
does preserve infima.
Proposition 3.2 There exists a Scott-continuous interval function which is neither bicontinuous nor Moore-continuous.
Proof Let S : IR −→ IR be the function defined by
⎧
if 0 < a
⎨ [1, 1],
if a ≤ 0 ≤ b
S([a, b]) = [−1, 1],
⎩
[−1, −1], if b < 0
(7)
called interval sign function. In what follows, we will prove that S is Scott-continuous but it
is not bi-continuous.
S is Scott-continuous: Let be a directed set of intervals and [u, v] =
. If 0 < u,
then
S
= [1, 1],
(8)
for each [a, b] ∈ , a ≤ u ≤ v ≤ b and there is an interval [c, d] ∈ such that 0 < c. Since
S() = {S([a,
S() = {[1, 1]}
b]) : [a, b] ∈ } then either
or S() = {[−1, 1], [1, 1]}, in
both case S() = [1, 1]. So, by Eq. (8), S() = S( ).
The case v < 0 is analogous.
If u ≤ 0 ≤ v then
S
= [−1, 1],
(9)
and necessarily for all [a, b] ∈ , 0 ∈ [a, b]. Hence, S() = {[−1, 1]} and
S() =
[−1, 1].
Therefore, in any case S() = S( ), i.e., S is Scott-continuous.
S does not preserve infima: Let = {[a, b] : 0 < a ≤ b < 2}. Clearly is a directed
set that = [0, 2] and hence S( ) = S([0, 2]) = [−1, 1]. Nevertheless, S() =
{S([a, b]) : 0 < a ≤ b < 2} = {[1, 1]} = [1, 1].
Thus, S( ) = S(), and therefore, it is not dE -continuous and so it is not bicontinuous.
S is
not Moore-continuous: Let X = [0, 0] and = 0.1. Then for any δ > 0 take Y =
δ δ
δ
2 , 2 . Clearly dM (X, Y ) = 2 < δ, still dM (S(X ), S(Y )) = dM ([−1, 1], [1, 1]) = 2 > .
Therefore, S is not Moore-continuous.
4 More formally F is bi-continuous if for every bi-directed set—let D = (D, ≤) be a poset. ⊆ D is
bi-directed if is a directed set of D and Dop = (D, ≥)—F( ) = F() and F( ) = F().
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B. Bedregal, R. Santiago
Corollary 3.3 DI − (DE ∪ DM ) = ∅.
Proposition 3.4 There exists a dE -continuous interval function which is neither bicontinuous nor Moore-continuous.
Proof Let T : IR −→ IR be the function defined by
[−1, 1], if a < 0
T ([a, b]) =
[0, 0], if a ≥ 0
(10)
We will prove that T is dE -continuous but it is not bi-continuous.
T preserves infima in IR: Let be a bi-directed set of intervals and [u, v] = . If u ≥ 0, then
(*) T ( ) = [0, 0], for each [a, b] ∈ , 0 ≤ u ≤ a ≤ b ≤ v and therefore T () = {[0, 0]}
and so T () = [0, 0]. So, by (*), T () = T (). If u < 0 then (**) T ( ) = [−1, 1],
for all [a, b] ∈ , u ≤ a ≤ b ≤ v and necessarily there exists [c, d] ∈ such that c < 0.
Hence, T () = {[−1, 1]} or T () = {[0, 0], [−1, 1]}, in both cases T () = [−1, 1]. So,
by (**), T ( ) = T ().
Therefore, in any case T () = T ( ), i.e., T is dE -continuous.
T does not preserve suprema: Let = {[a, 0] : −2 < a < 0}. Clearly isa directed
set
that
=
[0,
0]
and
hence
T
(
) = T ([0, 0]) = [0, 0]. Nevertheless,
T () =
{T ([a, 0]) : −2 < a < 0} = {[−1, 1]} = [−1, 1]. Thus, T ( ) =
T () and
therefore it is not Scott-continuous and so it is not bi-continuous.
Let X = [0, 0] and = 0.1. Then for any δ > 0 take Y =
T is not Moore-continuous:
δ
δ
− 2 , − 2 . Clearly dM (X, Y ) = 2δ < δ, still dM (T (X ), T (Y )) = dM ([0, 0], [−1, 1]) =
1 > . Therefore, T is not Moore-continuous.
Corollary 3.5 DE − (DI ∪ DM ) = ∅.
Proposition 3.6 There exists a Moore-continuous interval function which is neither dI continuous nor dE -continuous.
Proof The function F(X ) = m(X ) + 21 (X − m(X )) proposed in Moore (1979, p. 21) is
dM -continuous [the proof can be seen in Santiago et al. (2006) Proposition 3.1, p. 236] is not
inclusion monotonic and, therefore, is neither dI -continuous nor dE -continuous.
Corollary 3.7 DM − (DI ∪ DE ) = ∅.
Lemma 3.8 Let q A and q B be two quasi-metrics under the sets A and B, respectively. If
f : A −→ B is bi-continuous w.r.t. q A and q B , then f is continuous w.r.t. metrics q ∗A and
q B∗ .
Proof Since, f is continuous w.r.t. q A and q B , then ∀x∈A, ∀>0, ∃δ > 0 and ∃δ >
0, such that ∀y ∈ A, (q A (x, y) ≤ δ ⇒ q B ( f (x), f (y)) ≤ ) and (q A (x, y) ≤
δ ⇒ q B ( f (x), f (y)) ≤ ) So, defining δ ∗ = min(δ, δ) if q ∗A (x, y) ≤ δ ∗ then
max(q A (x, y), q A (x, y)) ≤ δ ∗ and therefore q A (x, y) ≤ δ ∗ and q A (x, y)) ≤ δ ∗ . Since δ ∗ ≤
δ and δ ∗ ≤ δ, q A (x, y) ≤ δ and q A (x, y) ≤ δ. So, since f is bi-continuous, q B ( f (x), f (y)) ≤
and q B ( f (x), f (y)) ≤ . Therefore, max(q B ( f (x), f (y)), q B ( f (x), f (y))) ≤ , i.e.,
q ∗ ( f (x), f (y)) ≤ . Hence f is continuous w.r.t. the metrics q ∗A and q B∗ .
Proposition 3.9 If F : IR −→ IR is bi-continuous, then F is Moore- and Scott-continuous.
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Proof As a consequence of Lemma 3.8 and Proposition 2.3, F is Moore-continuous. On the
other hand, by definition, F is Scott-continuous.
Corollary 3.10 (DE ∩ DI ) − DM = ∅.
Proposition 3.11 If F : IR → IR is Moore- and Scott-continuous, then F is bi-continuous.
Proof Suppose that F is Moore- and Scott-continuous, then by definition ∀X ∈ IR, ∀ > 0,
∃δM > 0, and ∃δI > 0 such that ∀Y ∈ IR, dM (X, Y ) < δM implies dM (F(X ), F(Y )) < and dI (X, Y ) < δI implies dI (F(X ), F(Y )) < .
Let X ∈ IR and > 0, then make δE = min{δ M , δI }.
Case 1: dM (X, Y )=dE (X, Y ). So, if dE (X, Y )<δE , then dM (X, Y )<δE ≤ δ M . Hence,
because F is Moore-continuous, dM (F(X ), F(Y ))<. Since, dE (F(X ), F(Y ))≤dM ((F(X ),
F(Y )), then dE (F(X ), F(Y ))<.
Case 2: dM (X, Y ) = dI (X, Y ), then dE (X, Y ) ≤ dI (X, Y ) (*).
• Case 2.1: Y ⊆ X , then dE (X, Y ) = 0 < δE . By monotonicity of F, F(Y ) ⊆ F(X ). So,
dE (F(X ), F(Y )) = 0 < .
• Case 2.2: X = [a, b], Y = [c, d] and a < c < b < d. Thus, by (*), | a − c |≥| b − d |,
then let Z = X ∩ Y = [c, b]. dE (X, Y ) = d − b = dE (Z , Y ) = dM (Z , Y ).
Thus, if dE (X, Y ) < δE then dE (Z , Y ) < δE . So, by case 1, dE (F(Z ), F(Y )) < (**). But, by monotonicity of F, F(Z ) ⊆ F(X ), dE (F(X ), F(Z )) = 0. Applying triangular inequality: dE (F(X ), F(Y )) ≤ dE (F(X ), F(Z )) + dE (F(Z ), F(Y )),
dE (F(X ), F(Y )) ≤ dE (F(Z ), F(Y )). Hence, by (**), dE (F(Z ), F(Y )) < .
• Case 2.3: X = [a, b], Y = [c, d] and a < b < c < d. Thus, by (*), | a − c |≥|
b − d |, let Z = [a, d]. dE (X, Y ) = d − b = dE (X, Z ) = dM (X, Z ). Thus, if
dE (X, Y ) < δE then dE (X, Z ) < δE . Hence, by case 1, dE (F(X ), F(Z )) < (***).
But, by monotonicity of F, F(Y )⊆F(X ), therefore dE (F(Z ), F(Z )) = 0. Applying triangular inequality: dE (F(X ), F(Y )) ≤ dE (F(X ), F(Z )) + dE (F(Z ), F(Y )),
dE (F(X ), F(Y )) ≤ dE (F(X ), F(Z )). So, by (***), dE (F(X ), F(Y )) < .
• Case 2.4 X = [a, b], Y = [c, d] and c < a < d < b is analogous to case 2.2.
• Case 2.5: X = [a, b], Y = [c, d] and c < d < a < b is analogous to case 2.3.
Corollary 3.12 (DM ∩ DI ) − DE = ∅.
Proposition 3.13 If F : IR → IR is Moore- and dE -continuous, then F is bi-continuous.
Proof The proof is analogous to that of Proposition 3.11.
Corollary 3.14 (DM ∩ DE ) − DI = ∅.
Corollary 3.15 DM ∩ DE = D M ∩ DI = DE ∩ DI = DM ∩ DE ∩ DI .
According to Corollaries 3.3, 3.5, 3.7, 3.10, 3.12 and 3.14, Fig. 1 shows the classification of interval functions with respect to various viewpoints of interval analysis, namely
the extensional, informational and metrical, i.e., DE , DI and DM , respectively. Particularly,
Corollary 3.15 establishes that the three viewpoints are captured by only two continuities,
and therefore, from now we make an abuse of language where bi-continuity will mean tricontinuity.
Now, with these established relations between the classes DM , DI and DE , it is suitable
to make up the following questions: (1) What is the relation of real continuous functions
with the functions in each class? (2) When we specify a real continuous function (using any
specification language), where can we classify its interval counterpart? (3) Moreover how is
this interval counterpart? Theorem 4.5, in the next section, will give us the answer.
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B. Bedregal, R. Santiago
Fig. 1 Classification of the various interval continuous functions
4 Relating correctness and optimality with number, extensional, and informational
viewpoints
Santiago et al. (2006) introduces the notion of interval representation and canonical interval representation, as a functional counterpart of correct and optimal interval algorithms,
respectively. The same reference makes an analysis of the relation between continuity of a
real function with Scott- and Moore-continuity of their canonical interval representations;
in that sense it was proved that optimality preserves continuity. In this section, we will
establish the relation between canonical interval representations of real continuous functions
with bi-continuous functions answering questions (1)–(3). But before, from Definition 4.1
to Proposition 4.4, we revisit the notion of interval representations.
Definition 4.1 Let f : R −→ R and F : IR −→ IR be real and interval functions,
respectively. F represents f , if for each X ∈ IR and x ∈ X , f (x) ∈ F(X ). The set of all
real functions represented by F is Rep(F) = { f : F represents f }.
As we mentioned before, the concept of representation captures exactly that of correctness.
But, it is suitable to ask if this concept can be applied to any function. Assuming Moore
arithmetics, the result of every interval function is an interval with real endpoints, meaning
that there are real functions which do not have representations. For example, the total real
function:
0, if x ≤ 0
f (x) = 1
x , otherwise
does not have an interval representation of f . Since for any interval [a, b], such that a <
0 < b, f ([a, b]) is an unbounded set, and hence it is not contained in any closed interval
with real endpoints. More generally, this problem arises for any asymptotic function,5 since
Moore arithmetics obligates outputs with real endpoints. However, it is possible to overcome
this problem, using some extended arithmetics (see Hickey et al. 2001; Kahan 1968; Kearfott
5 For us, a real function f is asymptotic if for some interval [a, b], the set { f (x ) : a ≤ x ≤ b} has no
supremum or no infimum.
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Some continuity notions for interval functions
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1996). Thus, to be non-asymptotic is a necessary condition for a real function to be represented
by an interval function described in terms of Moore arithmetics. On the other hand, according
to Fundamental theorem, monotonicity is a sufficient condition for representation. Therefore,
since bi-continuous functions are monotone, they always represent some real functions.
Assuming Moore arithmetics, the canonical representation for a real function f is
CIR( f )(X ) = [min{ f (x) : x ∈ X }, max{ f (x) : x ∈ X }]
(11)
That is, CIR( f )(X ) returns the narrowest interval containing all the values f (x), such that
x ∈ X . Particularly, when f is continuous, CIR( f )(X ) = f (X ). CIR( f ) is the optimal
(possible) interval representation of f . We formalize this by the following proposition.
Lemma 4.2 If F G 6 and f ∈ Rep(G) then f ∈ Rep(F).
Proof The proof is straightforward.
Proposition 4.3 An interval function F represents a real function f iff F CIR( f ).7
Proof (⇒) Straightforward [see Santiago et al. (2006) Proposition 5.2] (⇐) Straightforward
from Lemma 4.2.
Proposition 4.4 (Representation theorem) f is continuous iff CIR( f ) is Moore-continuous
iff CIR( f ) is Scott-continuous.
Proof See Santiago et al. (2006).
4.1 dI , dE and dM -continuities
The next theorem is the central of this paper; it establishes the connection between the
concepts of dI -continuity, dE -continuity, and dM -continuity with the usual notion of Euclidean
continuity on real functions. It states that the corresponding canonical interval representation
of a real continuous function is at least bi-continuous. A practical consequence of that is the
standard semantics of specifications which describe the canonical interval representation of
a real continuous functions is an interval bi-continuous function, i.e., an interval function
which continuity reflects the three aspects of intervals.
Theorem 4.5 (Representation theorem for bi-continuity) f is continuous if and only if
CIR( f ) is bi-continuous.
Proof (⇒) If f is continuous, then by the representation theorem CIR( f ) is Scottcontinuous. On the other hand, we proved that CIR( f ) is continuous with respect to dE ,
i.e., it preserves infima (unions). The proof is analogous to that of supremum preservation of
Santiago et al. (2006). Therefore, since CIR( f ) is Scott-continuous, it is inclusion monotonic
and hence for each nesting sequence of intervals , CIR( f )() is also a sequence of this
kind and CIR( f )( ) is one of its lower bounds. Therefore, CIR( f )( ) ⊆ CIR( f )().
Thus, we only need to prove that CIR( f )()⊆CIR( f )( ). If y∈ CIR( f )() and since
= and CIR( f )() is a nesting
sequence, then there is [a, b] ∈ such that
y ∈ CIR( f )([a, b]). So, since [a, b] ⊆ , then y ∈ CIR( f )( ).
(⇐) Straightforward from Proposition 3.9 and Theorem 4.4.
6 For every interval A, F(A) G(A).
7 For every interval A, F(A) CIR( f )(A).
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B. Bedregal, R. Santiago
Corollary 4.6 f is continuous iff CIR( f ) is dI , dE , and dM -continuous.
The next results state that the class of bi-continuous functions is richer than the class of
canonical interval representations of continuous real functions.
Proposition 4.7 Not every interval function Moore- and Scott-continuous is a canonical
representation of some continuous function.
Proof Consider the function I : IR −→ IR defined by
I ([a, b]) = [a − 1, b + 1].
Obviously, for any real function g, I = CIR(g). We will prove that I is both Moore- and
Scott-continuous.
Moore-continuity: Let X ∈ IR, > 0 and δ = . Then, for each Y ∈ IR if dM (X, Y ) ≤ δ
we have that
dM (I (X ), I (Y )) = dM (X + [−1, 1], Y + [−1, 1])
= max{|(x − 1) − (y − 1)|, |(x − 1) − (y − 1)|}
= max{|x − y|, |x − y|}
= dM (X, Y )
≤
So, I is Moore-continuous.
Scott-continuity: Let be a directed set. Then as early said, = = [ {x : X ∈
}, {x : X ∈ }]. So,
I
=I
{x : X ∈ }, {x : X ∈ }
=
{x : X ∈ } − 1, {x : X ∈ } + 1
=
{x − 1 : X ∈ }, {x + 1 : X ∈ }
=
I ()
Corollary 4.8 Not every interval bi-continuous function is the canonical interval representation of a real continuous function.
Proof Using an analogous reasoning to prove Proposition 4.7, is possible to prove that I also
preserve infima and therefore that it is bi-continuous. But as said in that theorem, I is not a
canonical interval representation of any real function.
Corollary 4.9 (DE ∩ DI ∩ DM ) − {CIR( f ) : f is continuous} = ∅
Therefore, continuity with respect to information, extension and metrics is a broader
concept than just that of the preservation of real continuity (see Fig. 2). Then, those results
induce the following section.
123
Some continuity notions for interval functions
445
Fig. 2 Classification of interval continuous functions and the CIR of real continuous functions
5 Final remarks and future works
Santiago et al. (2006) introduced the concepts of interval representation and canonical interval representation to formalize the principles of correctness and optimality for interval algorithms. Santiago et al. (2006) uses the definition of canonical interval representations to make
a comparative study of real continuity and its relation with Scott- and Moore-continuity.
In this work, we went further in those investigations, now considering interval functions
which are continuous with respect to the conjugated quasi-metric considered in Santiago et al.
(2006). With this third topology, we establish the following classes of interval functions: DM
for Moore-continuous, DI for Scott-continuous functions and DE for continuous functions
with respect to the conjugated quasi-metric. One relation between them is that DM ∩ DI =
DM ∩ DE = DI ∩ DE (which was explicitly exposed in Corollary 3.15). A function in
this intersection is called bi-continuous interval function. We showed that the class of bicontinuous functions is a proper superset of the class of canonical interval representations of
real continuous functions (see Fig. 2).
All the results exposed here can be extended to Rn and IRn . As future works, the authors
propose the investigation of continuity which comes both from the concept of i-metrics
proposed by de Santana and Santiago (2013) and from interval metrics proposed by Trindade
et al. (2010).
Acknowledgments This work was partially supported by Brazilian Research Council (CNPq) under the
processes numbers 306876/2012-4, 307681/2012-2 and 480832/2011-0.
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