Relative Computability and Uniform Continuity of Relations

A type-2 computable real function is necessarily continuous; and this remains true for relative, i.e. oracle-based computations. Conversely, by the Weierstrass Approximation Theorem, every continuous f:[0,1]->R is computable relative to some oracle. In their search for a similar topological characterization of relatively computable multivalued functions f:[0,1]=>R (aka relations), Brattka and Hertling (1994) have considered two notions: weak continuity (which is weaker than relative computability) and strong continuity (which is stronger than relative computability). Observing that uniform continuity plays a crucial role in the Weierstrass Theorem, we propose and compare several notions of uniform continuity for relations. Here, due to the additional quantification over values y in f(x), new ways of (linearly) ordering quantifiers arise, yet none of them turn out as satisfactory. We are thus led to a notion of uniform continuity based on the Henkin Quantifier; and prove it necessary for relative computability. In fact iterating this condition yields a strict hierarchy of notions each necessary, and the omega-th level also sufficient, for relative computability.


Introduction
A simple counting argument shows that not every (total) integer function f : N → N can be computable; on the other hand, each such function can be encoded into an oracle O ⊆ {0, 1} * that renders it relatively computable.Over real numbers, similarly, not every total f : [0, 1] → R can be computable for cardinality reasons; and this remains true for oracle machines.In fact it is folklore in Recursive Analysis that any function f computably mapping approximations of real numbers x to approximations of f (x) must necessarily be continuous; and the same remains true for oracle computations.Even more surprisingly, this implication can be reversed: If a (say, real) function f is continuous, then there exists an oracle which renders f computable § .This can for instance be concluded from the Weierstrass Approximation Theorem.A far reaching generalization from the reals to so-called admissibly represented spaces is the Kreitz-Weihrauch Theorem, cf.e.g.[Weih00,3.2.11] and compare the Myhill-Shepherdson Theorem in Domain Theory.The equivalence between continuity and relative computability has led Dana Scott to consider continuity as an approximation to computability.Now many computational problems are more naturally expressed as relations (i.e.multivalued) rather than as (single-valued) functions.For instance when diagonalizing a given real symmetric matrix, one is interested in some basis of eigenvectors, not a specific one.It is thus natural to consider computations which, given x, intensionally choose and output some value y ∈ f (x).Indeed, a multifunction may well be computable yet admit no continuous single-valued selection; cf.e.g.[Weih00, Exercise 5.1.13]or [Luck77].Hence multivaluedness avoids some of the topological restrictions of single-valued functions-but of course not all of them.Specifically it is easy to see that a multifunction f is relatively computable iff it admits a continuous so-called realizer, that is a function mapping any infinite binary string encoding some x to an infinite binary string encoding some y ∈ f (x).
However the single-valued case raises the hope for an intrinsic characterization of relative computability of f , without referral to Cantor space.Such an investigation has been pursued in [BrHe94], yielding both necessary and sufficient conditions for a relation to be computable relative to some oracle (which, there, is called relative continuity and we shall denote as relative computability).Brattka and Hertling have established what remains to-date the best counterpart to the Kreitz-Weihrauch Theorem for the multivalued case: Fact 1.Let X, Y be separable metric spaces and Y in addition complete.Then a pointwise closed relation f : X ⇉ Y is relatively computable iff it has a strongly continuous tightening ¶ Here, being pointwise closed means that f (x) := {y ∈ Y : (x, y) ∈ f } is a closed subset for every x ∈ X.We shall freely switch between the viewpoint of f :⊆ X ⇉ Y being a relation (f ⊆ X × Y ) and being a set-valued partial mapping f :⊆ X → 2 Y , x → f (x).Such f is considered total (written f : X ⇉ Y ) if dom(f ) := {x ∈ X : f (x) = ∅} coincides with X.Following [Weih08, Definition 7], g is said to tighten f (and f to loosen g) if both dom(f ) ⊆ dom(g) and ∀x ∈ dom(f ) : g(x) ⊆ f (x) hold; see Figure 1a) and note that tightening is obviously reflexive and transitive.Furthermore write f [S] := x∈S f (x) for S ⊆ X and range(f ) := f [X]; also f | S := f ∩ (S × Y ) and f | T := f ∩ (X × T ) for T ⊆ Y .Finally let f −1 := {(y, x) : (x, y) ∈ f } denote the inverse of f , i.e. such that (f −1 ) −1 = f and range(f ) = dom(f −1 ).

Continuity for Relations
For multivalued mappings, the literature knows a variety of easily confusable notions of continuity like [KlTh84,§7] or [ScNe07].Some of them capture the intuition that, upon input x, all y ∈ f (x) occur as output for some 'nondeterministic' choice [Brat03,Section 7]; or that the 'value' f (x) be produced extensionally as a set [Spre09].Here we pursue the original conception that, upon input x, some value y be output subject to the condition y ∈ f (x).§ It has been observed that a continuous function f : [0, 1] → [0, 1] will usually not have a least oracle rendering it computable [Mill04]   ¶ We reserve the original term "restriction" to denote either  1b): b) Call f weakly continuous if the following holds: c) Call f uniformly weakly continuous if the following holds: d) Call f nonuniformly weakly continuous if the following holds: e) Call f uniformly strongly continuous if the following holds: f ) Call f semi-uniformly strongly continuous if the following holds: Items a) and b) are quoted from [BrHe94, Definition 2.1].In the single-valued case, quantifications over y ∈ f (x) and y ′ ∈ f (x ′ ) drop out.Here, all a),b),d),f) collapse to classical continuity; and both c) and e) to uniform continuity.In the multivalued case, however, these notions are easily seen distinct.Note for instance that in f), δ may depend on x but not on y; whereas y may depend on ε in c) but not in b).Logical connections between the various notions are collected in the following Lemma 3. a) Strong continuity implies weak continuity b) but not vice versa.c) Weak continuity implies nonuniform weak continuity.d) Uniform weak continuity implies nonuniform weak continuity.e) Let f be uniformly weakly continuous and suppose that f (x) ⊆ Y is compact for every x ∈ X.Then f is weakly continuous.f ) Uniform strong continuity implies semi-uniform strong continuity which in turn implies strong continuity.g) For compact dom(f ) ⊆ X, nonuniform weak continuity implies uniform weak continuity.
Note that the (classically trivial) implication from (weak) uniform continuity to (weak) continuity in e) is based on the (again, classically trivial) hypothesis that f (x) ⊆ Y be compact.
Similarly, the classical fact that continuity on a compact set classically yields uniform continuity is generalized in g)+c).4d).e) Fix x ∈ dom(f ).By hypothesis there exists, to every ε = 1/n, some δ n and y n ∈ f (x) with: ).Now since f (x) is compact, there some subsequence y nm of y n converges to, say, y 0 ∈ f (x) with d(y nm , y 0 ) ≤ 1/m.We claim that this y 0 (which does not depend on ε anymore) satisfies Indeed, to arbitrary x ′ ∈ B(x, δ nm ) ∩ dom(f ), the hypothesis yields some y ′ ∈ B(y, 1/m) ∩ f (x ′ ).Then, by triangle inequality, it follows y ′ ∈ B(y 0 , 2/m).Note that a different x may require a different subsequence n m ; hence δ may become dependent on x even if it did not before.

Continuity and Computability of Relations
Recall that (relative) computability of a multifunction f :⊆ R ⇉ R means that some (oracle) Turing machine can, upon input of any sequence of integer fractions for every m ∈ N and some y ∈ f (x).More generally, a multifunction f :⊆ A ⇉ B between represented spaces (A, α) and (B, β) is considered (relatively) computable if it admits a (relatively) computable (α, β)-realizer, that is a function Lemma 5. Define the composition of multifunction f :⊆ X ⇉ Y and g :⊆ Y ⇉ Z as holds and both f and g are compact, then so is g • f .d) If range(f ) ⊆ dom(g) holds and if both f and g map compact sets to compact sets, then so does g • f .e) Fix representations α for X and β for Y .A multifunction Motivated by f), let us call a multifunction F as in e) an (α, β)-multirealizer of f .
The above notion composition for relations is, like that of 'tightening', from [Weih08, Section 3].Mapping compact sets to compact sets is a property which turns out useful below.It includes both compact relations (Lemma 3k) and continuous functions: Indeed, the signed digit representation ρ sd is well-known proper [Weih00, pp.209-210], i.e. preimages of compact sets are compact.Focusing on complete separable metric spaces and pointwise compact multifunctions, strong continuity is in view of Fact 1 (in general strictly) stronger than relative computability; whereas weak continuity is (again in general strictly) weaker than relative computability: {1} , is computable but not strongly continuous.
Proof.a) by contradiction: Suppose some oracle machine M computes this relation.On input of the rational sequence (0, 0, 0, . ..) as a ρ-name of x := 0 it thus outputs a ρ-name of y = 0, i.e. a rational sequence (p m ) with |p m | < 2 −m .In particular it prints p 1 > −1/2 after having read only finitely many elements from the input sequence; say, up to the (N − 1)-st element.Now consider the behavior of M on the input sequence (0, 0, . . ., 0, 2 −N , 2 −N , . ..) as ρ-name of x ′ := 2 −N : Its output sequence (p ′ m ) will, again, begin with for all m and for one of y = 0 =: ∋ y 1−j for the unique j ∈ {0, 1} with y = y j and is printed upon reading only the first, say, N ′ ≥ N elements of (0, 0, . . ., 2 −N , 2 −N , . ..).Finally it is easy to extend this finite sequence to a ρ-name of some x ′′ close to x ′ with y j ∈ g(x ′′ ) ∋ y 1−j ; and upon this input M will now, again, output elements p ′ 1 , . . ., p ′ N +1 which, however, cannot be extended to a ρ-name of any y ′′ ∈ g(x ′′ ): contradiction.b) see [BrHe94,p.24 and note that it is open in dom(f ) because y ′ ∈ B(y, ε) requires y ′ = y.Hence U y = dom(f ) ∩ j∈N B(q j,y , 1/n j,y ) for certain n j,y ∈ N and q j,y from the fixed dense subset of X.Now consider an encoding of (names of) these q j,y and n j,y as oracle.Then, given x ∈ dom(f ), search for some (j, y) with x ∈ B(q j,y , 1/n j,y ) ⊆ U y : when found, such y by construction belongs to f (x) and, conversely, weak continuity asserts x to belong to U y for some y.

Motivation for Uniform Continuity
Many proofs of uncomputability of relations or of topological lower bounds [Zieg09] apply weak continuity as a necessary condition: merely necessary, in view of the above example, and thus of limited applicability.The rest of this work thus explores topological conditions stronger than weak continuity yet necessary for relative computability.
Uniform continuity of functions is such a stronger notion -and an important concept of its own in mathematical analysis -yet does not straightforwardly (or at least not unanimously) extend to multifunctions.Guided by the equivalence between uniform continuity and relative computability for functions with compact graph, our aim is a topological characterization of oracle-computable compact real relations.One such characterization is Fact 1; however we would like to avoid (second-order) quantifying over tightenings.
To this end observe that every (relatively) computable function f is (relatively) effectively locally uniformly continuous [Weih00, Theorem 6.2.7], that is, uniformly continuous on every compact subset K ⊆ dom(f ) [KrWe87]: This suggests to look for related concepts for multifunctions, i.e.where δ does not depend on x.Uniform weak continuity in the sense of Definition 2c), however, fails to strengthen weak continuity because it allows y to depend on ε.

Henkin-Continuity
In view of the above discussion, we seek for an order on the four quantifiers ∀x ∈ dom(f ), ∃y ∈ f (x), ∀ε > 0, ∃δ > 0 such that y does not depend on ε and δ does not depend on x.This cannot be expressed in classical first-order logic and has spurred the introduction of the non-classical so-called Henkin Quantifier [Vaan07] Q H (x, y, ε, δ) = ∀x ∃y ∀ε ∃δ where the suggestive writing indicates that very condition: that y may depend on x but not on ε while δ may depend on ε but not on x.We thus adopt from [Bees85, p.380] the following Definition 9.Call f Henkin-continuous if the following holds: Observe that uniform strong continuity implies Henkin-continuity; from which in turn follows both weak continuity and uniform weak continuity.In fact, Henkin-continuity is strictly stronger than the latter two: Example 10. a) The relation g from Examples 4d) and 7a) is (compact and both weakly continuous and uniformly weakly continuous but) not Henkin-continuous.
b) It does, however, satisfy c) The relations from Examples 4b) and 7c) are (computable and) Henkin-continuous.
Its generalization from metric to uniform spaces is immediate but beyond our purpose.

Further Examples and Some Properties
Recall that, for single-valued functions, Henkin-continuity coincides with uniform continuity.
Therefore the first k − 1 elements of (q n ), and in particular the first k − 1 symbols of σ, extend to a ρ C -name τ of x ′ ; i.e. such that d(σ, τ ) < ε. f) Modifying the the argument at index (n, m) affects the image at index n, m ≥ n + m, i.e. the metric at weight ≤ 2 −(n+m) .⊓ ⊔ A classical property both of continuity and uniform continuity is closure under restriction and under composition.Also Henkin-continuity passes these (appropriately generalized) sanity checks: Observation 12. a) Let f :⊆ X × Y be Henkin-continuous and tighten g :⊆ X × Y .Then Proof.a) For g loosening f and in the definition of Henkin-continuity of g, the universal quantifiers range over a subset, and the existential quantifiers range over a superset, of those in the definition of Henkin-continuity of f .b) By hypothesis, we have ∀ε > 0 ∃δ > 0 ∀y ∈ dom(g) ∃z ∈ g(y) ∀y ′ ∈ B(y, δ) ∩ dom(g) ∃z ′ ∈ B(z, ε) ∩ g(y ′ ) (5) Thus, to ε > 0, take δ > 0 according to Equation (5) and in turn γ > 0 according to Equation (6).Similarly, to x ∈ dom(g •f ) ⊆ dom(f ), take y ∈ f (x) ⊆ dom(g) according to Equations ( 6) and (3); and in turn z ∈ g(y) according to Equation (5).This z thus belongs to g • f (x) and was obtained independently of ε, nor does γ depend on x.Moreover to The following further example in Item b) turns out as rather useful: with no consecutive digit pair 11 nor 11 nor 1 1 nor 11. b) For k ∈ N, each |x| ≤ 2 3 • 2 −k admits such an expansion with a n = 0 for all n ≤ k.And, conversely, x = ∞ n=k+1 a n 2 −n with (a n , a n+1 ) ∈ {10, 10, 01, 0 1, 00} for every be a signed digit expansion and k ∈ N such that (a n , a n+1 ) ∈ {10, 10, 01, 0 1, 00} for each n > k.Then every x d) Let Σ := {0, 1, 1, .}.The inverse ρ −1 sd : R ⇉ Σ ω of the signed digit representation is Henkin-continuous.
Proof.a) Start with an arbitrary signed digit expansion (a n ) of x and replace, starting from the most significant digits, i) any occurrence of 011 with 10 1, ii) any occurrence of 0 11 with 101, iii) any occurrence of 01 1 with 001, iv) any occurrence of 0 11 with 00 1.
Note that these substitutions do not affect the value ∞ n=−N a n 2 −n .Moreover the above four cases are the only possible involving one of 11 or 11 or 1 1 or 11 because, by induction hypothesis and proceeding from left (most significant) to right, no such combination was left before of the current position.On the other hand, rewriting Rule i) may well introduce a new occurrence of 11 before the current position; this is illustrated in the example of 0101011.Similarly for 11 in Rule ii).Therefore, we apply the rules in two loops: • An infinite outer one for n = −N, . . ., 0, 1, 2, . .., maintaining that neither 11 nor 11 nor 1 1 nor 11 occurs before position n • one application of rules i) to iv) to remove a possible occurrence at position n • followed by a finite inner loop for j running from n back to −N , iteratively removing occurrences which may have been newly introduced at position j.Observe that, after each termination of the inner loop, no occurrence remains before or at position n.Hence the process converges and yields an equivalent signed digit expansion with the desired property.b) Shifting/scaling reduces to the case k = 0; and negation to the case x > 0.
2 3 = 0.1010 . . . is an expansion with the claimed properties.So turn to 0 < x < 2 3 and, indirectly, w.l.o.g.suppose a 0 = 1.Extend this to a signed digit expansion of least value ∞ n=0 a n 2 −n = x with no consecutive 11, 11 , 1 1, 11. Due to monotonicity, this is attained by including digit 1 whenever admissible, namely 1.0 10 1 . . . of value x = 2 3 : a contradiction.For the converse, similarly observe that 0.1010 . . .has the largest value among all signed digit expansions with the claimed properties; and its value is 2 3 .c) Let x ′′ := k n=−N a n 2 −n and observe that x − x ′′ = ∞ n=k+1 a n 2 −n is by hypothesis a signed digit expansion satisfying (a n , a n+1 ) ∈ {10, 10, 01, 0 1, 00} for all n ≥ k + 1, hence 0 admits a signed digit expansion (possibly using combinations like 11) ] encoding the signed digit expansion (a n ) of x according to a).Due to c), every x ′ ∈ B(x, δ) ⊆ B(x, •2 −(k−1) /3) admits a signed digit expansion (b n ) coinciding with (a n ) for all n ≤ k − 1.Since every ρ sd -name includes the binary separator symbol, an appropriate name σ′ encoding (b n ) agrees with σ for at least the first k + 1 symbols, i.e. has distance at most 2 −k ≤ ε.

Other Characterizations and Tools
Let us call a mapping λ : N → N a modulus; and say that a multifunction f :⊆ X ⇉ Y is λ-continuous in (x, y) ∈ f if, to every m ∈ N and every x ′ ∈ dom(f )∩ B(x, 2 −λ(m) ) there exists some y ′ ∈ f (x ′ ) ∩ B(y, 2 −m ).Here, B(x, r) := {x ′ ∈ X : d(x, x ′ ) ≤ r} denotes the closed ball of radius r around x. Now Skolemization of "∀ε > 0∃δ > 0" yields Observation 14.A multifunction f :⊆ X ⇉ Y is Henkin-continuous iff there exists a modulus λ such that, for every x ∈ dom(f ), there exists y ∈ f (x) such that f is λ-continuous in (x, y); equivalently: if, for every x ∈ dom(f ), f admits some single-valued total selection f x : X → Y λ-continuous in x, f x (x) (but possibly not continuous anywhere else, see Example 16 below).
of multifunctions equicontinuous if they share a common modulus in the sense that the following holds: depicted in Figure 3 is compact and 1-Lipschitz.Moreover, f is computable but has no locally continuous selection in x 0 = 0.

Concerning Example 16b), the ratio min{|y
Note that proceeding from alphabet Σ to {0, 1} 2 affects the Lipschitz constant by a factor of 2.
(hence f (x ′ ) = {2 −k }, i.e. y = 2 −k ).Moreover every (x, y) ∈ f satisfies x/3 ≤ y ≤ x.Thus the following algorithm computes f : Given ] and proceed to interval #n + 1, otherwise switch to outputting the constant sequence [2 −n , 2 −n ].Note that for x = 0, the output sequence [a n /3, b n ] will indeed converge to y = 0.In case 3 holds and will result in the output of Proposition 17. a) I denote an ordinal and f i :⊆ X ⇉ Y (i ∈ I) an equicontinuous family of pointwise compact multifunctions and decreasing in the sense that f j tightens f i whenever j > i.Then f (x) := i:f i (x) =∅ f i (x) is again pointwise compact and Henkincontinuous a tightening of each f i .Moreover, if all f i are λ-continuous, then so is f .b) Let f : X ⇉ Y be λ-continuous and pointwise compact for some modulus λ.Then f has a minimal λ-continuous pointwise compact tightening.
Proof.a) Since the case of a finite I is trivial, it suffices to treat the case I = N of a sequence; the general case then follows by transfinite induction.Let x ∈ dom(f i ).Then f j (x) ⊆ f i (x) for each j > i, and hence f (x) = j≥i f j (x) ⊆ f i (x) is (compact and) the intersection of non-empty compact decreasing sets: f (x) = ∅, x ∈ dom(f ).Moreover let ε > 0 be arbitrary and consider an appropriate δ according to Equation (9) independent of x; similarly take y j ∈ f j (x) independent of ε as asserted by equicontinuity.Then the sequence (y j ) j>i belongs to compact f j (x) and thus has some accumulation point y ∈ f j (x) ⊆ f i (x) for each j: thus yields y ∈ f (x) independent of ε.W.l.o.g y j → y by proceeding to a subsequence.Now let d(x, x ′ ) ≤ δ.Then by hypothesis there exists y ′ j ∈ f j (x ′ ) with d(y j , y ′ j ) ≤ ε; and, again, an appropriate subsequence of (y ′ j ) converges to some y ′ ∈ f (x ′ ).Moreover, d(y, y ′ ) ≤ d(y, y j ) + d(y j , y ′ j ) + d(y ′ j , y ′ ) ≤ d(y, y j ) + ε + d(y ′ , y ′ j ) → ε. b) Consider the family F of all λ-continuous and pointwise compact tightenings of f .According to a), these form a directed complete partial order (dcpo) with respect to total restriction.More explicitly, apply Zorn's Lemma to get a maximal chain (f i ), i ∈ I. Then a) asserts that g(x) := i:f i (x) =∅ f i (x) defines a λ-continuous and pointwise compact tightening of f .In fact a minimal one: If h ∈ F tightens g, then h = f j for some j ∈ I because of the maximality of (f i ) i∈I ; hence g tightens f j .⊓ ⊔

Relative Computability requires Henkin-Continuity
With the above examples and tools, it is now easy to establish Theorem 18.Let K ⊆ R be compact.
a) If f : K ⇉ R is computable relative to some oracle, then it is Henkin-continuous.b) More precisely suppose F :⊆ {0, 1} ω ⇉ {0, 1} ω is a Henkin-continuous (ρ sd , ρ sd )-multirealizer of f : K ⇉ R (recall Lemma 5) which maps compact sets to compact sets.Then f itself must be Henkin-continuous, too; and has a Henkin-continuous tightening g : K ⇉ R mapping compact sets to compact sets.c) Conversely, if f : K ⇉ R is Henkin-continuous and maps compact sets to compact sets, then a Henkin-continuous (ρ sd , ρ sd )-multirealizer of f which maps compact sets to compact sets.

Henkin-Continuity does not imply Relative Computability
The relation from Example 4c) is Henkin-continuous but not relatively computable.On the other hand, it violates the natural condition of (pointwise) compactness.Instead, we modify Example 16 to obtain (counter-) ] is compact, total, and 1-Lipschitz (hence Henkincontinuous), but not relatively computable; see Figure 4.
Proof.Both f + and f − are closed and bounded and total.Moreover, the restriction f is 1-Lipschitz: To x ≤ 0 set y := 0 and δ := ε (1-Lipschitz); now if x ′ ≤ 0, y ′ := 0 will do; and if 0 < x ′ < δ, consider n ∈ N with 1/(n + 1) is 1-Lipschitz; hence f 1 is 1-Lipschitz-but not relatively computable: Given a name of x = 0, the putative realizer has the choice of producing either a name of y + = 0 or of y − = 1: knowing x only up to some δ = 1/n, n ∈ N. In the first case, i.e. already tied to f + , switch to an input x ′ := 1/(n + 1): clearly a point of discontinuity of f + .A similar contradiction arises in the second case.

Examples and Properties
Note that δ 2 in Equation (10), although independent of x 2 , may well depend on x 1 : which perhaps does not entirely express what might be expected from a notion of uniform continuity for relations.On the other hand, just like continuity on a compact set is in the single-valued case equivalent to uniform continuity, we establish Lemma 23.For compact X, total f : X ⇉ Y , and ℓ ∈ N, the following are equivalent: iii) There exists a total function λ : N → N such that For non-compact X, it still holds 'i)⇐ii)⇔iii)".
We call λ as in iii) a modulus of ℓ-fold Henkin-continuity of f .
Note also that equivalence of the Cauchy representation ρ to the signed digit representation ρ sd means that its inverse ρ −1 sd : R ⇉ Σ ω be computable.Hence Fact 1 asserts that ρ −1 sd has a strongly continuous (and w.l.o.g.pointwise compact) tightening.We now strengthen this as well as Proposition 13c)+d):

Conclusion
We have proposed a hierarchy of notions of uniform continuity for real relations based on the Henkin quantifier; and shown its ω-th level to characterize relative computability in the compact case.Our condition may be considered descriptionally simpler than the previous characterization from [BrHe94].Indeed, although Equation (13) does employ countably infinitary logic, Fact 1 even quantifies over subsets of uncountable R.
Question 30.Does Theorem 28 extend from compact subsets K of R d to general compact metric spaces?
A promising candidate replacement for ρ d sd K is provided in [BdBP10, Proposition 4.1].But is its inverse ω-fold Henkin-continuous (or does even admit a uniformly strongly continuous tightening) ?

Fig. 1 .
Fig.1.a) For a relation g (dark gray) to tighten f (light gray) means no more freedom (yet the possibility) to choose some y ∈ g(x) than to choose some y ∈ f (x) (whenever possible).b) Illustrating ǫ-δ-continuity in (x, y) for a relation (black)

Fig. 2 .
Fig.2.a) Example of a uniformly weakly continuous but not weakly continuous relation.b) A semi-uniformly strongly continuous relation which is not uniformly strongly continuous.c) A compact, weakly and uniformly weakly continuous relation which is not computable relative to any oracle.
Example 6. a) Let f : X → Y be a single-valued continuous function.Then f maps compact sets to compact sets.b) The inverse (ρ d sd ) −1 of the d-dimensional signed digit representation maps compact set to compact sets.c) The functions id : x → x and sgn : R → {−1, 0, 1} both map compact sets to compact sets; however their Cartesian product id × sgn does not map compact {(x, x) : −1 ≤ x ≤ 1} to a compact set.
Example 7. a) The relation (2) from Example 4d) is not computable relative to any oracle.b) The relation from Example 4c) is (uniformly strongly continuous but, lacking pointwise compactness) not computable relative to any oracle.c) The closure of the relation from Example 4b), that is with graph

Fig. 4 .
Fig. 4. A compact total 1-Lipschitz but not relatively computable relation.(Dashed lines indicate alignment and are not part of the graph)