150

数理解析研究所講究録
第2083巻 2018年 150-155

On a certain discontinuous mapping
as a continuous relation(summary)
Satoru Yoshida

Hitoshi Frusawa

Tottori University ofKagoshima University

Environmental Studies

1Introduction

In classical mathematics, the following mapping F is a mapping of \mathbb{R} into \{0 , 1 \},and is discontinuous at 0 :F(x):=\left\{ begin{ar ay}{l 0\mathrm{i}\mathrm{f}x\leq0\ 1\mathrm{i}\mathrm{f}x>0. \end{ar ay}\right.
In Bishop’s constructive mathematics (BISH) [2], we cannot prove that such adiscontinuous mapping exists, since, in this framework, the existence implies a

independent principle LPO (the least limited principle of Omniscience). Also,in Weihrauch’s computable mathematics, F is not computable, and therefore

is not continuous (see [1] and [9])However, [1] gives some investigations ofclassical discontinuous mappings in the view point as relation, and does notionsof continuous relations.

But we have not known whcther the existence of adiscontinuous mappiiig is justified by recognising it as a continuous re ation inBISH. This paper gives an investigation of continuous relation in BISH.
Constructive mathematics is formalized in intuitionistic logic, and always
requires a constructive proof, i.e. a proof which can be regarded as an algorithm.

Bishop has given in [2] an informal framework of constructive mathematics,

which is a proper subsystem of classical mathematics. Actually, it does notinclude an axiom that is refused in classical mathematics, and therefore its
constructive proof is acceptable in classical mathematics. On the other hand,some classical theorems are independent on BISH. For example, the following

property, called LPO (the least limited principle of Omniscience):\forall\{a_{n}\}\in\{0, 1\}^{\mathrm{N}}[\forall n(a_{n}=0)\vee\exists n(a_{n}=1)],
holds classically, but cannot be proved in BISH, since we cannot give an algo‐rithm which decides whether a_{n}=0 for all natural number n , or find a natural
1 . But the negation of LPO does not hold; that is, LPO
number n with a_{n}=151

is independent on BISH. Moreover it is proved in BISH that LPO is equivalent
to the following:\forall x\in \mathbb{R}[x>0\vee x\leq 0],
where \mathbb{R} is the set of all Cauchy reals. We therefore cannot decide whether
a given real number is either positive or non‐positive. However the following

apartness property holds in BISH (see [2, Corollary in p.24], [4, proposition(5.3)], etc\forall a, b, c\in \mathbb{R}[a<b\rightarrow a<b\vee b<c].
This is proved by comparison with two rational Cauchy sequence. This property
has an important role, and also is used in the proof of Theorem 9.
Section 2 gives a constructive investigation of continuous properties givenin [1]. In Section 3, we consider some continuity properties of a mapping for

further study of continuous relations

2Continuity properties of a relation

Let \mathbb{R} denote a Euclid space of all Cauchy reals, X, Y and V subsets of \mathbb{R}, R
a relation on X\times \mathrm{Y} i.e. a subset of X\times Y , and $\varepsilon$>0 . Remark that “‘ A\neq\emptyset ”
means that we can take an element of A . Let x and y be in \mathbb{R} . We now givesome notations as follows:[x]R:=\{y\in Y:(x, y)\in R\}, R[y]:=\{x\in X:(x,y)\in R\},[V]R:=\{y\in \mathrm{Y}:R[y]\cap V\neq\emptyset\},R[V]:=\{x\in X:[x]R\cap V\neq\emptyset\},dom(R) :=\{x\in X : \exists y\in Y((x, y)\in Rrange(R) :=\{y\in \mathrm{Y} : \exists x\in X((x, y)\in R

N(x, $\varepsilon$):=\{y\in \mathbb{R}:|x-y|< $\varepsilon$\}.

We now consider notions of continuity of relation, defined in [lj.
Definition 1.

1. Let (x,y) a point in X \times \mathrm{Y}. R is continuous at (x,y) if for each k in\mathbb{N} , there exists n in \mathrm{N} such that N_{Y}(y, 2^{-k}) \cap[x']R \neq \emptyset for all x' in

N_{X}(x, 2^{-n})\cap \mathrm{d}\mathrm{o}\mathrm{m}(R).

2. R is continuous if R is continuous at all point (x, y) in R.
3. R has a continuous restriction if there exists a continuous relation S\subset Rsuch that that \mathrm{d}\mathrm{o}\mathrm{m}(S)=\mathrm{d}\mathrm{o}\mathrm{m}(R) .

4. R is weakly continuous if for all x in \mathrm{d}\mathrm{o}\mathrm{m}(R) , there exists y in R[x] suchthat R is continuous at (x, y) .

It is easy to show the following proposition.

152

Proposition 1. In Definition 1, (2) \Rightarrow (3) \Rightarrow (4) holds.
We here define F as a relation.
F :=\{(x, y)\in \mathbb{R}\times\{0 , 1 \} : (x\leq 0\rightarrow y=0)\wedge(x>0\rightarrow y=1

Proposition 2.
1. F\subset \mathbb{R}\times \mathbb{R}.

\mathrm{d}\mathrm{o}\mathrm{m}(F)=\{x\in \mathbb{R}:x\leq 0\}\cup\{x\in \mathbb{R} : x>0\}.

2.

3. \mathbb{R}=\mathrm{d}\mathrm{o}\mathrm{m}(F) if and only if LPO holds.
4. range (F)=\{0 , 1 \}.

5.

N_{\{0,1\}}(0,2^{-k}) =\{0\} for all k.

6.

[2^{-k}]R=\{1\} for all k.

Theorem 3. F is not continuous at (0,0) , and hence is not weak continuous.

Proof.

N_{\{0,1\}}(0,2^{-k})\cap[2^{-(k+1)}]R=\emptyset for all k in \mathrm{N} by (5) and (6) of Proposi‐tion 2. That is, F is not continuous at (0,1) . Since (0,1)\not\in R, F is not weakly\squarecontinuous.

We now consider an analogy of F . Let $\varepsilon$>0 , and set
F_{ $\varepsilon$} :=\{(x, y) : (x< $\varepsilon$\rightarrow y=0)\wedge(x>0\rightarrow y=1

It is clear that Fõ is not a mapping but a relation and that \mathrm{d}\mathrm{o}\mathrm{m}(F_{ $\varepsilon$})=\mathbb{R} by
the apartness property.

Theorem 4. For any $\varepsilon$>0, F_{ $\varepsilon$} is continuous.

Proof. Fix any $\varepsilon$ > 0 . Let (x,y) be an element of Fõ, and any k in N. Then

0 < x or x < $\varepsilon$ .
2^{-n}<x .

In the former case, we haveThen, for any x' in N_{\mathbb{R}}(x, 2^{-n}) ,y= 1 .

Take some n in \mathbb{N} with

N_{\{0,1\}}(1,2^{-k})\cap[x']F_{ $\varepsilon$}=\{1\}.
In the latter case, we have y=0 . Take some n in \mathbb{N} with x+2^{n}<e , Then, forx' in N_{\mathbb{R}}(x, 2^{-n}) ,N_{\{0,1\}}(0,2^{-k})\cap[x']F_{ $\epsilon$}=\{0\}.
\squareWe next consider weaker continuity of a relation.
Definition 2.

1. Let (x, y) a point in X \times Y. R is left continuous at (x,y) if for each kin \mathbb{N} , there exists n in \mathrm{N} such that N_{Y}(y, 2^{-k})\cap[x']R\neq\emptyset for all x' inN_{X}(x, 2^{-n})\cap \mathrm{d}\mathrm{o}\mathrm{m}(R)\cap(-\infty, x).

153

2. R is left continuous if R is left continuous at all point (x, y) in R.
3. R has a left continuous restriction if there exists a left continuous relation
S\subset R such that \mathrm{d}\mathrm{o}\mathrm{m}(S)=\mathrm{d}\mathrm{o}\mathrm{m}(R) .
4. R is weakly left continuous if for all x in \mathrm{d}\mathrm{o}\mathrm{m}(R) , there exists y in R[x]such that R is left continuous at (x, y) .

Left continuity for a mapping is given in [5, p.57], [8, Sect \mathrm{B} , Ch. 508] andso on.

We clearly have the following proposition.

Proposition 5. In Defimtion2, (2) \Rightarrow (3) \Rightarrow (4) hold_{\mathcal{S}}.
In classical mathematics, F is also a left continuous mapping on \mathbb{R} . We

similarly obtain this matter in BISH as follows:
Theorem 6. F is left continuous.

Proof. Let (x, y) be any point of F , and any k in N. Then x in \mathrm{d}\mathrm{o}\mathrm{m}(F) , andtherefore x \leq 0 or 0 < x . In the former case, we take any n in \mathrm{N} , and have

N_{\{0,1\}}(0,2^{-k})\cap[x']F=\{0\} for all x' in N_{\mathbb{R}}(x, 2^{-n})\cap \mathrm{d}\mathrm{o}\mathrm{m}(F)\cap(-\infty, x) . In thelatter case, we can choose in \mathrm{N} with 2^{-n}<x , and then N_{\{0,1\}}(0,2^{-k})\cap[x']F=n\{1\} for all x' in N_{\mathbb{R}}(x, 2^{-n})\cap \mathrm{d}\mathrm{o}\mathrm{m}(F)\cap(-\infty,x) .

3\square

On continuity properties of a mapping

In this section, we consider some standard continuity properties of a mapping
for more investigation of a continuous relation. Let f be a mapping of a subset
X of \mathbb{R} into \mathbb{R} and x in X . We say that f is continuous at x if for each k in \mathrm{N},there exists N in \mathrm{N} such that, given any n\geq N,f(N_{X}(x, 2^{-n})) \subset N_{\mathbb{R}}(f(x), 2^{-k}),where f(X) denotes the image by f. f is continuous if f is continuous at allpoint X . It is easy to show that f is a continuous mapping if and only if it is acontinuous relation.

A subset A of \mathbb{R} is open in \mathbb{R} if for each x in A , there exists some $\delta$>0 suchthat N_{\mathbb{R}}(x, $\delta$)\subset A

We can easily show the following theorem.
Theorem 7. Let f be a mapping of \mathbb{R} into \mathbb{R} . Then f is continuous if and only
if, whenever O is an open subset of \mathbb{R} , then the inverse f^{-1}(O) is open in \mathbb{R}.
We next consider sequential continuity of a mapping. In BISH, a contin‐uous mapping of a metric space into a metrics is sequentially continuous, but

the converse cannot be proved (see [7] and so on). We therefore need to in‐vestigate sequential continuity of a mapping as another continuity property, forconsidering sequentially continuous relations.

154

Let A be a subset of \mathbb{R}. \overline{A} means the closure of A i.e.

\overline{A} :=\{x\in \mathbb{R} : \forall $\varepsilon$>0[N_{\mathbb{R} (x, $\varepsilon$)\cap A\neq\emptyset]\}A subset of \mathbb{R} is closure in \mathbb{R} if \overline{A}=A.

The following lemma is proved in the same way as Ishihara’s Trick ([3,Lemma 3.2.1] and [6, Lemma 1
Lemma 8. Let f be a mapping from \mathbb{R} into \mathbb{R}, \{x_{n}\} a sequence in \mathbb{R} and x areal number. Assume that f satisfies f(\overline{A}) \subset\overline{f(A)} for all subset A of \mathbb{R} . Then,for positive real numbers a and b with a<b, |f(x_{n})-f(x)|>a for some n , or|f(x_{n})-f(x)|<b for all n.
We finally show the equivalence between sequential continuity and anothercontinuity properties.

Theorem 9. Let f be a mapping from \mathbb{R} into \mathbb{R} .

Then the followings areequivalent.

1. f \dot{u} sequentially continuous i.e. f satisfies that for all x in \mathbb{R} and se‐quence \{x_{n}\} in \mathbb{R} , if \{x_{n}\} converges to x in \mathbb{R} , then the sequence { f (xn)}converges to f(x) in \mathbb{R}.

2. For all x in \mathbb{R} and \mathcal{S} equence { x_{n}\} in \mathbb{R} , if \{x_{n}\} converges to x in \mathbb{R} , thenthere exists a subsequence \{f(x_{n_{k}})\} of\backslash { f (xn)} converging to f(x) in \mathbb{R}.
3. For all subset Cof\mathbb{R}, if C is closed in \mathbb{R} , then the subset f^{-1}(C) is closed
in \mathbb{R}.

4. For all subset A of \mathbb{R}, f(\overline{A}) \subset\overline{f(A)}.

We here note the most important part of the proof of (4) \Rightarrow (1). Let x
be a real number, and \{x_{n}\} a sequence converging to x in \mathbb{R} . Take a strictlyincreasing sequence \{N_{k}\} in \mathrm{N} such that |x_{n}-x| <2^{-k} for any n\geq N_{k} . Let $\varepsilon$
be any positive number. By Lemma 8, construct an increasing binary sequence\{$\lambda$_{k}\} such that$\lambda$_{k}=0 \displaystyle \Rightar ow \foral i\leq k[\exists n\geq N_{i}|f(x_{n})-f(x)|>\frac{ $\epsilon$}{2}]$\lambda$_{k}=1 \Rightarrow \exists i\leq k[\forall n\geq N_{i}|f(x_{n})-f(x)|< $\epsilon$].
Then the assumption (4) implies $\lambda$_{k}=1 for some k.

Acknowledgement. This work was supported by Tottori University of Envi‐ronmental Studies Grant‐in‐Aid for Special Research. The authors are also
thankful for the Research Institute for Mathematical Sciences, a Joint Us‐

age/Research Center located in Kyoto University.

155References[1] V.Brattka and P. Hertling: Continuity and computability of relation, In‐formatik berihite 164, Universität in Hagen, 1994.

[2] E. Bishop, Foundations of constructive analysis, Ishi Press, 2012.

[3] Douglas S. Bridges and Luminita Simona Vita: Techniques of ConstructiveAnalysis, Springer, 2006.

[4] Douglas Bridges and Fred Richman: Varieties of Constructive Mathemat‐ics, Cambridge University Press, 1987.

[5] I. N. Bronshtein, K. A. Semendyayev, G. Musiol and H. Muehlig: Handbookof Mathematics fifth editton, Springer, 2007.

[6] Hajime Ishihara: Continuity and nondiscontinuity in constructive mathe‐matics, J. Symbolic Logic 56 (1992), pp.1349‐1354.
[7] Hajime Ishihara: Sequential continuity in constructive mathematics, In:C.S. Calude, M.J. Dinneen and S. Sburlan eds., Combinatorics, Com‐putability and Logic, Proceedings of the Third International Conference
on Combinatorics, Computability and Logic, (DMTCS’OI) in Constanga
Romania, July 2‐6, 2001, Springer‐Verlag, London.

[8] The Mathematical Society of Japan:Japanese), Iwanami, 2007.

Suugaku‐Jiten 4th edition (in[9] Klaus Weihrauch: Computable Analysis, Springer, 2000.
Satoru Yoshida

Tottori University of Environmental Studies
1‐1‐1 Wakabada‐kita, Tottori 689‐1111
Japan.
satoru‐y@kankyou‐u.ac.jp
Hitoshi Furusawa

Kagoshima University
1‐21‐35 Korimoto, Kagoshima 890‐0065
Japan
furusawa@sci.kagoshima -\mathrm{u} .ac.jp