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Between continuous and uniformly continuousfunctions on Rn ∗Krzysztof CiesielskiDepartment of Mathematics, West Virginia University,Morgantown, WV [email protected] page: http://www.math.wvu.edu/homepages/kciesDikran DikranjanDipartimento di Matematica e Informatica, Universit`a di UdineVia delle Scienze 206, 33100 Udine, [email protected] study classes of continuous functions on Rnthat can be approx-imated in various degree by uniformly continuous ones (uniformly ap-proachable functions). It was proved in [BDP1] that no polynomial func-tion can distinguish between them. We construct examples that distin-guish these classes (answering a question from [BDP1]) and we oﬀer ap-propriate forms of uniform approachability that enable us to obtain ageneral theorem on coincidence in the class of all continuous functions.1 IntroductionOur set theoretical and topological notations are standard and follow [Ci] and [E],respectively. Given a metric space X we denote by C(X) (or simply C) the setof continuous functions f: X → R. We use the abbreviation “u.c.” for “uni-formly continuous.” The class of uniformly continuous functions (from currentlyconsidered space X into R) will be denoted by UC. The main classes studiedin this paper are the following.∗Work partially supported by the NATO Collaborative Research Grant CRG 950347.AMS classiﬁcation numbers: Primary 54C30, 41A30; Secondary 54E35, 54B30, 26A15.Key words and phrases: metric spaces, uniformly continuous maps, uniform approachabil-ity, truncation, Cantor function.1Deﬁnition 1.1 [BD] Let X be a metric (or, more generally, uniform) space,f: X → R, K ⊆ X, and M ⊆ X.1. g: X → R is a K, M-approximation of f if g is u.c., g[M ] ⊆ f[M], andg(x)=f(x) for each x ∈ K.2. f is uniformly approachable (brieﬂy, UA) if f has a K, M-approximationfor each compact K ⊆ X and each M ⊆ X.3. f is weakly uniformly approachable (brieﬂy, WUA) if f has an x, M-approximation (that is, more formally, {x},M-approximation) for eachx ∈ X and for each M ⊆ X.Clearly every u.c. function is UA, and WUA is a special case of UA when thecompact set K reduces to a point x. It is also not diﬃcult to check that everyWUA function is continuous [BD, fact 2.2]. Thus UC → UA → WUA → C.This justiﬁes the title of the paper.Is should be also mentioned here that for the functions from R to R threeof the above notions coincide, that is, UA ↔ WUA ↔ C. (See [BD, prop. 3.5].)However Maxim R. Burke noticed [BD, example 3.3] that on R2there arecontinuous non-WUA functions. (In fact, f : R2→ R, f (x, y)=xy, is such afunction.) Let us recall that WUA functions were introduced in [DP] under thename “uniformly approachable functions” (see also [B]). They provided an easyand elegant solution of the problem of whether the uniform continuity can becharacterized (in appropriate sense) by means of closure operators in the senseof [DT] (since WUA functions are easily seen to be continuous with respect toevery closure operator).It is easy to see that if the set M is empty then K, M-approximationsalways exist and the notion is uninteresting. (For K = ∅ any u.c. extension g off|K to a u.c. function, which exists by Katˇetov extension theorem, is a K, ∅-approximation of f.) However, if M is properly chosen, then the conditiong[M] ⊆ f [M ] is much stronger than it could be expected. In fact, it has beenproved in [BD, thm. 8.5] that, under the continuum hypothesis CH, for everyseparable metric space X there exists a set M ⊂ X, called a magic set, suchthat any ∅,M-approximation g of a nowhere constant function f must bea truncation of f , that is, g must be constant on each connected componentof {x ∈ X: f(x) = g(x)}. This motivates the introduction of the class TUAof truncation-UA functions, that is, functions f ∈ C(X) such that for everycompact set K ⊆ X there is a u.c. truncation g of f which coincides with f onK. Clearly TUA → C for every locally compact space X. The result quotedabove shows that, under CH, UA → TUA for nowhere constant functions onevery separable metric space X.(TakeaK, M-approximation of the constantfunction f with respect to a magic set M.) Since the TUA functions have asimpler geometrical description, this stimulated the further study of the magicsets and their properties and lead to a deep investigation of the question whether2the existence of magic sets can be proved without the assumption of CH ([BD,Question 14.1]). After some preliminary negative results (see [BC1, BC2]),Shelah and the ﬁrst named author showed that this cannot be done even for thereals R [CS].In the comparison of TUA and UA in separable metric spaces (and in partic-ular, in Rn), Berarducci, Pelant and the second named author [BDP1] noticedrecently that uniform approachability provides also a good connection to prop-erties of the functions related to ﬁbers. A function f : Rn→ R has distantconnected components of ﬁbers (brieﬂy, DCF ) if any two connected compo-nents of distinct ﬁbers f−1(x) and f−1(y) are at positive distance. They proved[BDP1, cor. 6.20] that for the functions on Rnone hasUA → WUA → TUA ↔ DCF.They also proved that UA ↔ WUA ↔ TUA for all polynomial functions fromRnto R and, more generally, for all functions with ﬁbers having ﬁnitely manyconnected components. The following question was left open in [BDP1, Question8.2(1)]:Question 1.2 Do the properties UA, WUA, and TUA coincide for all contin-uous functions Rn→ R?Also, the strength of the condition g[M] ⊆ f[M ] suggested that the diﬀerencebetween UA and WUA is very small. In fact, the following open problem wasraised in [BD]:Question 1.3 Let X be a connected metric space and let f: X → R be a WUAfunction. Is then f also UA?In this paper we will answer negatively these questions. More precisely, wegive contributions mainly in three directions:(1) We answer negatively Question 1.2 by constructing a function f ∈ C(R2)which shows that, in Rnwith n ≥ 2, TUA does not imply even WUA.This shows that UA and WUA are too strong conditions to participatein a set of equivalent conditions containing TUA and DCF. This moti-vated us to introduce here the following weaker version of UA: a functionf: X → R is UAd(densely uniformly approachable) if it admits uniformK, M-approximations for every dense set M and for every compact setK. One can deﬁne