# A MEAN VALUE THEOREM FOR METRIC SPACES

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1 A MEAN VALUE THEOREM FOR METRIC SPACES PAULO M. CARVALHO NETO 1 AND PAULO A. LIBONI FILHO 2 Abstract. We present a form of the Mean Value Theorem (MVT) for a continuous function f between metric spaces, connecting it with the possibility to choose the ε δ(ε) relation of f in a homeomorphic way. We also compare our formulation of the MVT with the classic one when the metric spaces are open subsets of Banach spaces. As a consequence, we derive a version of the Mean Value Propriety for measure spaces that also possesses a compatible metric structure. Keywords metric space, mean value property, mean value theorem, homeomorphism, average of functions. MSC (2010) Primary: 30L99, 54E40; Secondary: 26A15, 26E Introduction One of the most important results in Analysis is the Mean Value Theorem (MVT), which is used to prove many other significant results, from Several Complex Variables to Partial Differential Equations. It claims, in its original formulation (see Dieudonné, [5]) that if I = [a,b] R is a closed real interval and B is a Banach space, then every pair of continuous functions f : I B and φ : I R with derivatives in (a,b) satisfies the following implication (1) f (ξ) B φ (ξ)for all ξ (a,b) = f(b) f(a) B φ(b) φ(a). The most known consequence of the previous result happens when we fix some constant M > 0 and set φ(ξ) = M(ξ a). This guarantees that (2) f (ξ) B M for all ξ (a,b) = f(b) f(a) B φ(b) φ(a) = M(b a). There is also the classic form of the MVT for functions of several variables, which is directly obtained from the previous version. Consider B 1,B 2 Banach spaces and U B 1 an open subset. If a,b U are such that the line segment [a,b] U and f : U B 2 is a continuous function in [a,b] which is differentiable in (a,b), then we achieve a similar conclusion of (2) by considering f composed with the curve t a+(b a)t,where 0 t 1. Inspired by Dieudonné, several authors generalized such theorems to more abstract spaces and/or to less regular functions. For instance, Clarke and Ledyaev in [1, 2] proposed the study of MVT in Hilbert spaces for lower semi-continuous functions in a new multidirectional sense. In a subsequent paper, Radulescu and Clarke (cf. [12]) proved that in certain particular spaces the locally Lipschitz functions also fulfill a variant of MVT. Another distinct and celebrated approach was obtained by D. Preiss in [11]. He considered an Asplund space B and a Lipschitz continuous function f : B R. By denoting as D the set of 1 The author has been partially supported by FAPESP. Process 2013/ The author has been partially supported by CAPES. Process PNPD 2770/

2 2 P. M. CARVALHO NETO AND P. A. LIBONI FILHO points where f is differentiable, D. Preiss proved that D is dense in B and then concluded that for any x,y B the following MVT holds f(x) f(y) L x y B, wherel = sup f (x) L(B,R). x D An important formulation is due to M. Turinici, which proved (cf. [13]) a version resembling the original MVT in (1) to partially ordered metric spaces under some restrictive conditions on the metric. As we can see, there is a rich literature claiming different formulations of the MVT. On the other hand, there is a lack of discussion for a version of the theorem which is suitable for metric spaces. In what follows, inspired by Turinici s paper and collected work about this topic, we propose a lot of results that discuss the viability of MVT for those general spaces. To do this, we must clarify what a suitable substitute of MVT means when no differential and/or vectorial framework is assumed. To motivate our definition, let us examine the situation pictured in (2). Let (x r,x+r) (a,b) and consider Ψ : [0,r) R + R + given as (3) Ψ(d) = sup f (ξ) B. ξ [x d,x+d] If we write d R to indicate the Euclidian distance in R, then it follows that the MVT in (2) is equivalent to (4) f(x) f(y) B Ψ(d R (x,y))d R (x,y)for ally (x r,x+r). Note that the previous formulation is expressed in a metric fashion, with the aid of a certain function Ψ. Also, observe that extra proprieties of Ψ can be obtained from some regularity that f may possess. This discussion motivate us to consider the next definition. Definition 1. Let (M 1,d 1 ) and (M 2,d 2 ) be metric spaces. Consider a function f : M 1 M 2 and a point x M 1. We say that f satisfies the Mean Value Inequality (MVI) in the open ball B M1 (x,r) if there is a function Ψ : [0,r) R + R + such that d 2 (f(x),f(y)) Ψ(d 1 (x,y))d 1 (x,y) for all y B M1 (x,r). Also, the following properties must hold (i) Ψ is a continuous function; (ii) The function I : [0,r) R + R + defined as I(d) = Ψ(d)d is a non decreasing homeomorphism over its image. In this work we have two main objectives. First, establish a necessary and sufficient condition for the MVI in the open ball B M1 (x,r). Note that our applications will be defined in abstract metric spaces, not necessarily partially ordered ones (cf. [13]). To deal with this problem, we introduce a subclass of continuous applications that, roughly speaking, are the ones which we can find a ε δ(ε) relation that possesses some regularity. Our second goal is to show that the MVI in a Banach space is reduced to the supremum of the norm of the derivative in a open ball. We point out that in metric spaces there is no usual way to define directions, as opposed to what can be done in Banach spaces. Therefore, it is natural to expect that in our metric-only situation, the supremum is taken in the open ball, instead of the line segment.

3 A MEAN VALUE THEOREM FOR METRIC SPACES 3 To conclude, this paper is organized in the following sequence: in Section 2, we discuss the basic properties associated with sets that describes the regularity of a given function defined on a metric space. These sets are fundamental to the base structure of Sections 3 to 5, in which we investigate the ε δ(ε) regularity of continuous functions. In Section 6, we establish a necessary and sufficient condition for the MVI. Further, in Section 7, we derive a version of the Mean Value Propriety, which is about computing and estimating average of functions, and how it relates to its counterpart in Harmonic Analysis. 2. Preliminary Ideas and Initial Concepts Throughout this paper, (M 1,d 1 ) and (M 2,d 2 ) denote metric spaces, unless stated otherwise. For i = 1,2, we write B Mi (x,r) to represent the open ball centered at x M i with radius r > 0. Also, let F(M 1,M 2 ) (or simply F) be the set of all functions between M 1 and M 2. An element of F M 1 R + \{0} is called a triplet. We say that a positive real number δ is suitable for the (f,x,ε) triplet if f(b M1 (x,δ)) B M2 (f(x),ε). In other words, δ > 0 is suitable for the (f,x,ε) triplet if the following implication holds y M 1 and d 1 (x,y) < δ = d 2 (f(x),f(y)) < ε. With the previous definition, it is clear that f : M 1 M 2 is continuous at x M 1 if, and only if, for every ε > 0 there exists a suitable number for the (f,x,ε) triplet. Based in the notion introduced above, we now discuss properties of some new structures. Definition 2. Given the (f, x, ε) triplet, consider the set f,x (ε) = {δ > 0 : δ is suitable for the (f,x,ε) triplet}. The idea behind the previous set, is to capture all possible choices of δ > 0 one can find while trying to prove that a certain function is continuous. Note that we are not excluding the possibility of f,x (ε) =, which would means that f is discontinuous at x. It is important to understand every possible scenario for the set of suitable numbers for a triplet. Theorem 3. Let (f,x,ε) be a triplet. One, and only one of the following alternatives occurs. (i) f,x (ε) = ; (ii) f,x (ε) = (0, ); (iii) There exists a certain δ > 0 such that f,x (ε) = (0,δ]. Proof. Let us first prove that f,x is connected. We claim that if δ 2 f,x (ε) and if 0 < δ 1 δ 2, then δ 1 f,x (ε). Indeed, if y M 1 and d 1 (x,y) < δ 1, then d 1 (x,y) < δ 2. Since δ 2 is suitable for the (f,x,ε) triplet, we conclude that d 2 (f(x),f(y)) < ε. This guarantees that δ 1 is also suitable for the (f,x,ε) triplet. Now let us check that f,x is closed from the right side. More precisely, if {δ n } n=1 f,x (ε) is an increasing sequence that converges to δ, then δ f,x (ε). Fix y M 1 such that d 1 (x,y) < δ. Choose a natural number N such that δ δ N < δ d 1 (x,y). Since the sequence is increasing, we conclude that d 1 (x,y) < δ N. But we already know that δ N is suitable for the (f,x,ε) triplet, therefore we conclude that d 2 (f(x),f(y)) < ε. Once y M 1 was an arbitrary choice, δ is also suitable for the (f,x,ε) triplet. Now, we are able to conclude the argument. Three alternatives occur. (i) f,x (ε) =. Then, we are done;

4 4 P. M. CARVALHO NETO AND P. A. LIBONI FILHO (ii) f,x (ε) is nonempty unbounded. Then, by the hereditary property above, f,x (ε) = (0, ); (iii) f,x (ε) is nonempty bounded. Then, by the sequential argument we just developed, f,x (ε) = (0,δ], where δ = sup f,x (ε) = (0,δ]. To study more properties of f,x, fix x M 1 and consider the family { f,x (ε) : ε > 0}. Roughly speaking, we are going to show that the set of suitable numbers for the (f,x,ε) triplet does not became smaller as ε gets bigger. Proposition 4. Consider any function f : M 1 M 2. For each element x M 1, if 0 < ε 1 ε 2, then f,x (ε 1 ) f,x (ε 2 ). Proof. If f,x (ε 1 ) =, there is nothing to be done. On the other hand, suppose that there exists a value δ f,x (ε 1 ). Hence, for y M 1 with d 1 (x,y) < δ, we have that d 2 (f(x),f(y)) < ε 1. But ε 1 ε 2, therefore for any y M 1 with d 1 (x,y) < δ, we have that d 2 (f(x),f(y)) < ε 2. In other words, δ f,x (ε 2 ). Definition 5. Given x M 1 and a function f : M 1 M 2, define E f (x) = {ε > 0 : f,x (ε) is a non empty bounded set}. Shortly, we may say that E f (x) captures all the possible values of ε > 0 such that there exists a maximum real number δ > 0 that verifies the sentence y M 1 and d 1 (x,y) < δ = d 2 (f(x),f(y)) < ε. Working in the same way as we did in Theorem 3, we examine the full range of topological possibilities about the set E f (x). Note that, from now on, the continuity of the function f at x plays an important role. This allow us to state and prove the following result. Theorem 6. Let f : M 1 M 2 be a given function which is continuous at x M 1. Then one, and only one of the following alternatives occurs. (i) If f is an unbounded, then E f (x) = (0, ); (ii) If f is a constant function, then E f (x) = ; (iii) In the other cases, there exists some real number ε > 0 such that E f (x) = (0,ε] or E f (x) = (0,ε). Proof. Note that since f is continuous at x M 1, Proposition 4 guarantees that E f (x) is an interval. Now let us prove that f is a constant function if, and only if, E f (x) =. Indeed, if there exists x M 1 such that E f (x) =, then for any ε > 0 we have that f,x (ε) = (0, ). Thus, for any ε,δ > 0 we must have the following y M 1 and d 1 (x,y) < δ = d 2 (f(x),f(y)) < ε. Therefore, for any y M 1 and any ε > 0 we have that d 2 (f(x),f(y)) < ε; which implies that for any y M 1, d 2 (f(x),f(y)) = 0. The other part of the claim is trivial. To conclude, let us prove that E f (x) is a proper subset of (0, ) if, and only if, f is bounded. Indeed, let f be bounded. In this case, there exists ε > 0 such that f(m 1 ) B M2 (f(x),ε). In other words, for any δ > 0 f(b M1 (x,δ)) B M2 (f(x),ε).

5 A MEAN VALUE THEOREM FOR METRIC SPACES 5 This last statement implies that any δ is suitable for the (f,x,ε) triplet and therefore f,x (ε) = (0, ), which means that ε / E f (x). If there exists ε (0, ) \ E f (x), by definition f,x (ε) = (0, ). Hence, for any y M 1 we have that d 2 (f(x),f(y)) < ε, which implies that f is bounded. Example 7. Note that all the situations listed in the last case can happen, as we can see in the following couple examples. (i) Consider M 1 = M 2 = R and d 1 = d 2 the real Euclidean metric. If f : M 1 M 2 is the characteristic function of [0, ), then (0, x ], ε (0,1] and x 0 (0, ), ε > 1 and x 0 f,x (ε) =, ε (0,1] and x = 0 (0, ), ε > 1 and x = 0 which implies that E f (0) = and E f (x) = (0,1] for x 0; (ii) Let M 1 = M 2 = [0, ) and d 1 = d 2 be the induced real Euclidean metric. If f : M 1 M 2 is defined as f(x) = 1 e x, then ( ] 0,ln 1 1 ε, ε (0,1) f,0 (ε) = ( 0, ), ε 1. Therefore E f (0) = (0,1). Until this moment, the constructions that we made do not seem to have a direct connection with the original function f. However, now that we have discussed these prerequisites, we are ready to define the function that is related to the continuity of f. Definition 8. Given f : M 1 M 2 and x M 1 such that E f (x), define Π f,x : E f (x) (0, ) as Remark 9. A few remarks are in order. Π f,x (ε) = max f,x (ε). (i) Since the set f,x (ε) admits a maximum for every ε E f (x), the function Π x is well defined; (ii) Is a direct consequence, from the previous section, that Π x is a non decreasing function that for every ε E f (x) provided the largest possible number such that f(b M1 (x,π f,x (ε))) B M2 (f(x),ε); (iii) Even if, during the construction of the function Π f,x, the original function f plays an important role, whenever there is no possibility of confusion, we omit f as an index, writing instead just Π x. 3. The Continuity of Π x We now focus our attention to the continuity properties of the function Π x. Lemma 10. Let f : M 1 M 2 be a non constant and continuous function. Choose x M 1 and suppose that for any r > 0 the closure of B M1 (x,r) is compact in M 1. Under these conditions, for all ε E f (x) the following holds.

6 6 P. M. CARVALHO NETO AND P. A. LIBONI FILHO (i) If {ε n } n=1 E f (x) is an increasing sequence that converges to ε, then lim max f,x(ε n ) = max f,x (ε); n (ii) If {ε n } n=1 E f (x) is a decreasing sequence that converges to ε, then lim max f,x(ε n ) = max f,x (ε). n Proof. We only verify the first part; the second one follows in a similar way. By Proposition 4, the map ε max f,x (ε) is non decreasing. Denote, for simplicity δ = lim n max f,x(ε n ) and δ = max f,x (ε) If δ δ, then the non decreasing property above gives δ < δ. Consider the auxiliary sequence {δ n } n=1 given by δ n = δ [ ] +δ δ δ + 2 2(n+1) and observe that it satisfies the following properties (i) δ n ( δ +δ)/2; (ii) δ < δ n < δ. Since for each n = {1,2,...} the number δ n is strictly bigger then δ, we have that δ n f,x (ε n ). Therefore, for each n, there exists y n M 1 such that d 1 (y n,x) < δ n and d 2 (f(y n ),f(x)) ε n. Since y n B M1 (x,δ), choosing a subsequence if necessary, we can assume that there exists y B M1 (x,δ) such that y n y. Then, when n, we obtain d 1 (y,x) ( δ +δ)/2 and d 2 (f(y),f(x)) ε, and conclude that δ f,x (ε), which is a contradiction. Theorem 11. Let f : M 1 M 2 be a non constant and continuous function. Choose x M 1 and suppose that for any r > 0 the closure of B M1 (x,r) is compact in M 1. Under these conditions Π x is a continuous function. Proof. Choose and fix ε 0 E f (x). Lets prove the equality lim Π x (ε) = Π x (ε 0 ) = lim Π x (ε). ε ε 0 t ε + 0 We shall begin proving the left limit. Let { ε n } be sequence to the left of ε 0 such that ε n ε 0. It is not difficult to construct another increasing sequence {ε n } with the following proprieties (i) ε n < ε n, for each n N; (ii) ε n ε 0, when n. Then we know that f,x (ε n ) f,x ( ε n ) f,x (ε 0 ) for all natural number n. Using the maximum function in this sequence of inclusions, we deduce max f,x (ε n ) max f,x ( ε n ) max f,x (ε 0 ). Finally, applying the limit when n on both sides and using Lemma 10 we obtain max f,x (ε 0 ) lim n max f,x( ε n ) max f,x (ε 0 ). Since { ε n } was an arbitrary sequence, we conclude the proof of the left limit. The right one is obtained in a similar way and therefore we are done.

7 A MEAN VALUE THEOREM FOR METRIC SPACES 7 The next step in our work is to extend the domain of Π x. To this end, we recall a useful definition. Definition 12. A function f : M 1 M 2 is called locally constant at a point x M 1, if there is exists r > 0 such that f BM1 (x,r) is a constant function. Observe that locally constant functions are almost similar to the constant functions, when we compute its Π x function. Hence, we shall exclude this class of functions in our future results. Theorem 13. Let f : M 1 M 2 be a continuous function. Choose x M 1 and suppose that f is not locally constant at x. In this case, 0 is an accumulation point of E f (x) and lim Π x (ε) = 0. ε 0 + Proof. Since f is continuous and non constant, it follows easily (Theorem 6) that 0 is an accumulation point of E f (x). In fact, E f (x) turns out to be an interval in this case. Now, consider {ε n } E f (x) a decreasing sequence that converges to 0. By monotonicity, we have that f,x (ε n+1 ) f,x (ε n ). If diama is the diameter of the set A, then diam f,x (ε n+1 ) diam f,x (ε n ). Since Π x (ε) = max f,x (ε) = diam f,x (ε), we observe that ( ) lim Π x(ε n ) = diam f,x (ε n ). n If δ n=1 f,x (ε n ), then f(b(x,δ)) B(f(x),ε n ) for any n N. Since ε n 0, we conclude that f is locally constant at x. Since by hypothesis this situation cannot happen, we obtain that n=1 f,x (ε n ) =. Therefore lim n Π x (ε n ) = 0. Following the same final steps done in Theorem 11, we conclude the proof of this theorem. This last theorem allow us to continuously extend our definition of Π x to ε = 0 if f is not locally constant at x. This extension will be useful in the next result, which connects the image of Π x with the set of suitable values. Theorem 14. Let f : M 1 M 2 be a continuous function. Choose x M 1 such that f is not locally constant at x. Suppose that for any r > 0 the closure of B M1 (x,r) is compact in M 1. Under these conditions, for all ε E f (x) we have that f,x (ε) = Π x (0,ε]. Proof. If δ > 0 is suitable for the (f,x,ε) triplet, then Π x (0) < δ Π x (ε). Since Π x is continuous, it must exist some ε (0,ε] such that Π x (ε ) = δ. Conversely, if ε (0,ε], then f(b M1 (x,π x (ε ))) B M2 (f(x),ε ) what guarantees that f(b M1 (x,π x (ε ))) B M2 (f(x),ε). This proves that Π x (ε ) is suitable for the (f,x,ε) triplet. n=1 4. Π x as a Homeomorphism There is no special reason to expect that Π x is a homeomorphism. For example, if R is considered with the discrete metric, than any given f : R R is a continuous function. In the case where f(x) = x, it is quite easy to see that 1 is the maximal suitable value for the (f,x,ε) triplet. More precisely, we have that Π x (ε) = 1 for all ε (0,1). Inspired by this example, we shall now investigate under which circumstances we can guarantee that Π x is a homeomorphism. We start with some basic definition. Definition 15. Given a function f : M 1 M 2 and a point x M 1, we say that f is adherent at x if the following holds

9 A MEAN VALUE THEOREM FOR METRIC SPACES 9 Without loss of generality, assume that there exists ỹ ε such that lim n y n,ε = ỹ ε. Using a limit argument, we have that for all ε [ε 1,ε 2 ], there is some ỹ ε M 1 such that d 1 (x,ỹ ε ) δ and d 2 (f(x),f(ỹ ε )) ε. If we suppose, for an instant, that d 1 (x,ỹ ε ) < δ = Π x (ε), a directly consequence would be that d 2 (f(x),f(ỹ ε )) < ε, which is a contradiction. Because of that, for all ε [ε 1,ε 2 ], there is some ỹ ε M 1 such that d 1 (x,ỹ ε ) = δ and d 2 (f(x),f(ỹ ε )) ε. Since f is adherent at x, we conclude that ε [ε 1,ε 2 ], ỹ ε M 1 such that d 1 (x,ỹ ε ) = Π x (ε) and d 2 (f(x),f(ỹ ε )) = ε. In other words, there exists ỹ ε2 M 1 such that d 1 (x,ỹ ε2 ) = Π x (ε 2 ) and d 2 (f(x),f(ỹ ε2 )) = ε 2. Since f is adherent at x and d 1 (x,ỹ ε2 ) = Π x (ε 1 ), we conclude that ε 2 = d 2 (f(x),f(ỹ ε2 )) ε 1 what implies that ε 1 = ε 2. Therefore, Π x is an injective function. Remark 18. Since any continuous and injective function g : I R, defined on any interval I R, is a homeomorphism (from I to g(i)), this last theorem guarantees that function Π x is a homeomorphism. In other words, it guarantees that the dependence ε δ(ε) from the continuity of a function in a metric space can be described by a homeomorphism. Next theorem specifies more clearly which functions can be expected to be adherent. From now on, if (M 1,d 1 ) denotes a metric space, we write B M1 [x,r] to denote the closed ball B M1 [x,r] = {y M 1 : d 1 (x,y) r}. Theorem 19. Let f : M 1 M 2 be continuous at x M 1. Suppose that M 1 and M 2 are metric spaces such that the closure of the ball B Mi (x,r) is the closed ball B Mi [x,r]. Then f is adherent at x. Proof. Letε E f (x). Noticethatf(B M1 (x,π x (ε)) B M2 (f(x),ε). Indeed,letp f(b M1 (x,π x (ε)). Then, there exists z B M1 (x,π x (ε)) such that f(z) = p. On the other hand, z is the limit of some sequence (z n ) of elements in B M1 (x,π x (ε)). Since f is a continuous function, we must have that p = limf(z n ). In other words, p f(b M1 (x,π x (ε)). But f is a continuous map, therefore f(b M1 (x,π x (ε)) B M2 (f(x),ε) what guarantees that p B M2 (f(x),ε). Now, using the hypothesis about the closure of the balls, we conclude that f(b M1 [x,π x (ε)]) B M2 [f(x),ε]. At last, suppose that there exists y M 1 such that d 1 (x,y) = Π x (ε). It follows that y B M1 [x,π x (ε)], what guarantees that f(y) B M2 [f(x),ε]. This concludes the proof. We finish this section with an important theorem, that put together some previous results and discussions. Theorem 20 (Π x as a Homeomorphism). Suppose that f : M 1 M 2 is any given function, x M 1 and assume the following (i) f is continuous; (ii) f is adherent at x; (iii) f is not locally constant at x; (iv) For any r > 0 the closure of B M1 (x,r) is compact in M 1. Then there exists an interval I x and a function Π x : I x J x R + such that (i) 0 I x R + ; (ii) Π x is a monotonic increasing homeomorphism; (iii) f(b M1 (x,δ)) B M2 (f(x),ε) δ Π x (ε), for all ε I x ; (iv) Π x (0) = 0.

10 10 P. M. CARVALHO NETO AND P. A. LIBONI FILHO 5. Enclosed Functions At this point, we observe that for a certain class of functions (see Theorem 20) we have already managed to construct a homeomorphism that is related to the continuity properties of f. Definition 21. Given a function f : M 1 M 2 and a point x M 1, we say that f is strongly enclosed at x if f satisfies the conclusions (i), (ii), (iii) and (iv) of Theorem 20. Furthermore, the Π x map is called the strong continuity function of f at x. We can also say that f is enclosed at x if f satisfies all the previous conclusions, with the possible exception of the implication f(b M1 (x,δ)) B M2 (f(x),ε) = δ Π x (ε). In this case, the Π x map is just called a continuity function of f at x. If the function is strongly enclosed (or enclosed) at all points of the domain, we say that the function is strongly enclosed (or enclosed). Roughly speaking, we may not have a maximum suitable number for enclosed functions, while the opposite occurs with the strongly enclosed functions. In general, to guarantee that a function is strongly enclosed, we require some compactness hypothesis (see Theorem 20). On the other hand, all locally Lipschitz functions are enclosed. It is interesting to note that the strong continuity function associated with a strong enclosed function f is unique, where the uniqueness here is considered in the sense of germs at 0. Note that this does not happen in the enclosed case. We are now interested in finding other classes of function which are enclosed or strongly enclosed. Definition 22. Given a function f : M 1 M 2 and a point x M 1, we are going to say that f satisfies the Lagrange Propriety at x if the following two conditions hold (i) There exists a C 1 function Γ : ( ζ,ζ) R R such that Γ(r) = d 2 (f(x),f(y)) sup y B M1 [x,r]\{x} d 1 (x,y) for all r (0,ζ); (ii) For all r (0,ζ), there is an element y r B M1 [x,r]\{x} such that d 2 (f(x),f(y)) sup y B M1 [x,r]\{x} d 1 (x,y) = d 2(f(x),f(y r )). d 1 (x,y r ) Example 23. Consider M 1 = M 2 = R and d 1 = d 2 the real Euclidean metric. Assume that f : M 1 M 2 is given by f(x) = 1 e x. Since (1 e y )/y is a positive decreasing function, we have that Γ(r) = 1 e y sup = 1 er y B R [0,r]\{0} y r = d 2(f(0),f( r)). d 1 (0, r) In other words, f satisfies the Lagrange Propriety at 0 with Γ(r) = er 1 r. The truly importance of the Lagrange class is uncovered by the following result, which connects the Γ function with the strong enclosed definition. Theorem 24. Suppose that f : M 1 M 2 is any given function that satisfies the Lagrange Propriety at x M 1. If Γ(0) 0 and Γ (0) > 0, then f is strongly enclosed at x. Furthermore, the strong continuity function of f is a C k diffeomorphism, provided that Γ is a C k application. Proof. Let : ( ζ,ζ) R be given by (t) = tγ(t). Since is a C k application and (0) is an isomorphism, we can use the Inverse Function Theorem and assume that ( ζ1,ζ 1) : ( ζ 1,ζ 1 ) ( ζ 2,ζ 2 ) is a C k diffeomorphism.

11 A MEAN VALUE THEOREM FOR METRIC SPACES 11 Let 1 be the inverse map of ( ζ1,ζ 1). Then, 1 (ε)γ( 1 (ε)) = ε, which implies that 1 (ε) = ε sup y B M1 [x, 1 (ε)]\{x}. d 2 (f(x),f(y)) d 1 (x,y) We claim that 1 (ε) is suitable for (f,x,ε) triplet. Indeed, if d 1 (x,y) < 1 (ε), then This implies that d 2 (f(x),f(y)) d 2 (f(x),f(y)) < d 2 (f(x),f(y)) sup d 1 (x,y). y B M1 [x, 1 (ε)]\{x} d 1 (x,y) d 2 (f(x),f(y)) sup 1 (ε) = ε. y B M1 [x, 1 (ε)]\{x} d 1 (x,y) If ε is small enough, let us prove that δ = 1 (ε) is the maximum possible suitable number for (f,x,ε) triplet. Indeed, if δ = 1 (ε), then (δ) = ε, which implies that δγ(δ) = ε. Remember that for each δ, there exists an element y δ such that Γ(δ) = d 2(f(x),f(y δ )). d 1 (x,y δ ) Hence, δd 2 (f(x),f(y δ )) = εd 1 (x,y δ ). It is now clear that d 2 (f(x),f(y δ )) = ε if, and only if, d(x,y δ ) = δ. Since y δ B M1 [x,δ]\{x}, we also conclude that d 2 (f(x),f(y δ )) < ε if, and only if, d(x,y δ ) < δ. Now suppose that δ is not the maximum possible suitable number for (f,x,ε) triplet. Therefore, we can find some E > 0 such that Since y δ B M1 [x,δ]\{x}, we have that Making E 0, we then conclude that Therefore, as proved above, d 1 (x,y) < E +δ = d 2 (f(x),f(x)) < ε. d 1 (x,y δ ) < E +δ = d 2 (f(x),f(y δ )) < ε. d 1 (x,y δ ) δ = d 2 (f(x),f(y δ )) < ε. d 1 (x,y δ ) δ = d 2 (f(x),f(y δ )) < ε = d(x,y δ ) < δ. Because of the last inequality, we have that y δ B M1 [x,δ]\{x} actually satisfies d(x,y δ ) < δ. Since Γ (0) > 0 and Γ is a C 1 function, we know that Γ is non negative on a certain interval starting at the origin. Considering that ε is small enough, we have that Γ(d 1 (x,y δ )) < Γ(δ) = d 2(f(x),f(y δ ). d 1 (x,y δ ) Note that the last inequality is a contradiction, since by definition Γ(d 1 (x,y δ )) d 2(f(x),f(y δ ). d 1 (x,y δ ) Therefore δ = 1 (ε) is the maximum suitable number for the (f,x,ε) triplet. In other words, Π x = 1 is the strongly enclosed function associated to f. Since the function ( ζ1,ζ 1) : ( ζ 1,ζ 1 ) ( ζ 2,ζ 2 ) is a C k diffeomorphism, the proof is finished.

12 12 P. M. CARVALHO NETO AND P. A. LIBONI FILHO Corollary 25. Assuming the last theorem hypothesis, we also have that ε ε Π x (ε) = = d 2 (f(x),f(y)) Γ(ξ(ε)), d 1 (x,y) sup y B M1 [x,ξ(ε)]\{x} where ξ satisfies the following conditions ξ(0) = 0 ξ (εγ(ε)) = 1 Γ(ε)+εΓ (ε) Proof. We just have to note that ξ = Π x = 1. The rest of the conclusion follows from the Chain Rule. Example 26. If M 1 = M 2 = [0, ) and d 1 = d 2 are the real Euclidean metric, we know that if function f : M 1 M 2 is given by f(x) = 1 e x, then Π 0 (ε) = ln 1 ε2 ε+ 1 ε 2 +O(ε3 ). Now let us consider a slightly different situation. Suppose that M 1 = M 2 = R and d 1 = d 2 are the real Euclidean metric. Suppose that f : M 1 M 2 is given by f(x) = 1 e x. Since our domain is an open set, we can use the last theorem to find an expression for Π 0 (ε). We also know that f satisfies the Lagrange Propriety with Γ(r) = 1 er r. Under these conditions, ξ satisfies the following system ξ(0) = 0 ξ (εγ(ε)) = for sufficiently small ε. Therefore ξ(ε) = ln(1+ε) and 1 Γ(ε)+εΓ (ε) Π 0 (ε) = ln(1+ε) ε ε2 2 +O(ε3 ). It is noteworthy that a simple domain change can alter the Π continuity function expression. This phenomenon should not be surprising, since there is a connection between the ε δ(ε) relation of f and its domain. Corollary 27. Suppose that B 1 and B 2 are Banach spaces. In addition to the last theorem hypothesis, suppose that M 1 is an open set of B 1 and also assume that M 2 B 2. If f is differentiable and M is the maximum value of function t f (t) L(B1,B 2), then ε M Π x(ε) ε Γ(0) for all sufficiently small ε. If t f (t) L(B1,B 2) does not reach a maximum value, then the first inequality is reduced to 0 Π x (ε). 6. The Mean Value Inequality Now we are ready to present our main theorem. Theorem 28 (Mean Value Inequality for Metric Spaces). Given a function f : M 1 M 2 and a point x M 1, the following are equivalent (i) f is enclosed in x and Π 1 x is differentiable at 0 + ; (ii) f satisfies the MVI (see Definition 1) for metric spaces in some B M1 (x,r).

13 A MEAN VALUE THEOREM FOR METRIC SPACES 13 In addition, suppose that M 1 and M 2 are open sets contained in Banach spaces B 1 and B 2, respectively. If f is differentiable and strongly enclosed at x, then there exists R > 0 such that for any y B M1 (x,r). Ψ(d 1 (x,y)) sup f (t) L(B1,B 2) Proof. Suppose that f is enclosed at x and consider I x = D(Π x ). Define Ψ : J x R + by Π 1 x (d), if d J x \{0} Ψ(d) = d (Π 1 x ) (0 + ), if d = 0, where J x = Π x (I x ). Observe that Ψ is continuous and that the function d Ψ(d)d is a non decreasing homeomorphism. Now choose R > 0 with the following two properties holding. (i) R < (1/2)supJ x ; (ii) For any y B M1 (x,r), we have d 2 (f(x),f(y) < supi x. Lety B M1 (x,r). Ifε (0,R),thenthevalued 1 (x,y)+εisnotsuitableforthe(f,x,d 2 (f(x),f(y))) triplet. Therefore we obtain the following inequality Π x (d 2 (f(x),f(y))) < d 1 (x,y)+ε. Applying Π 1 x on both sides and taking the limit as ε 0, we achieve d 2 (f(x),f(y)) Π 1 x (d 1 (x,y)) = Ψ(d 1 (x,y))d 1 (x,y). Conversely, let us suppose that f satisfies the MVI. Then, there exists a function Ψ : [0,r) R + R + with the properties described on Definition 1. Set Π x as the inverse of the function d Ψ(d)d. Hence, if d 1 (x,y) < δ and δ Π x (ε), MVI guarantees that d 2 (f(x),f(y)) Ψ(d 1 (x,y))d 1 (x,y) = Π 1 x (d 1 (x,y)). Sinced 1 (x,y) < Π x (ε)andπ x isstrictlyincreasing, weobtainthatπ 1 x (d 1 (x,y)) < Π 1 x (Π x (ε)) = ε, i.e., d 2 (f(x),f(y)) < ε. Finally, to prove the last statement, assume that M 1 and M 2 are open sets contained in Banach spaces B 1 and B 2, respectively and f is differentiable. Choose R > 0 such that for any y B M1 (x,r) d 1 (x,y) sup f (t) L(B1,B 2) < supi x /2. Now let us prove that for any y B M1 (x,r), the value d 1 (x,y) is suitable for the (f,x,r) triplet, where the value r = d 1 (x,y) sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2). Indeed, if d 1 (x,ỹ) < d 1 (x,y), we have that d 2 (f(x),f(ỹ)) d 1 (x,ỹ) sup BM1 [x,d 1(x,ỹ)] f (t) L(B1,B 2) d 1 (x,ỹ) sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2) < d 1 (x,y) sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2). Therefore,sincef isstronglyenclosedatx,d 1 (x,y) Π x (d 1 (x,y) sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2) what guarantees Π 1 x (d 1 (x,y)) d 1 (x,y) sup f (t) L(B1,B 2), )

14 14 P. M. CARVALHO NETO AND P. A. LIBONI FILHO which finally implies that Ψ(d 1 (x,y)) sup f (t) L(B1,B 2), and the proof is complete. Notice that the differentiability requirement of Π 1 x at 0 is the natural substitute for f being differentiableonitsdomain. Alsonotethat,sinceΨisacontinuousfunctionandsup BM1 [x,d 1(x,y)] f (t) L(B1,B 2) may not be continuous, there is no hope to expect anything better than the inequality provided by the last theorem. Ontheotherhand,iff iscontinuouslydifferentiablewehavethaty sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2) defines a continuous function. Moreover, it provide us more precise information about the function Ψ, as we can verify in the following result. Theorem 29. In addition to the last theorem hypothesis, suppose that M 1 and M 2 are open sets contained in Banach spaces B 1 and B 2, respectively. If f is continuously differentiable and if there is a direction v B 1 such that f (x) v B2 = f (x) L(B1,B 2) 0, then there exists R > 0 such that for all y B M1 (x,r). Ψ(d 1 (x,y)) = Proof. Choose R > 0 with the following two properties sup f (t) L(B1,B 2) (i) d 1 (x,y)sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2) < supi x, for any y B M1 (x,r); (ii) R < (1/2)supJ x. where I x and J x where defined on Theorem 28. Let y B M1 (x,r). Suppose that for any ε (0,R), the positive value d 1 (x,y) + ε is not suitable for the (f,x,r) triplet, where r = d 1 (x,y)sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2). Under these circumstances, we obtain ) Π x (d 1 (x,y) sup f (t) L(B1,B 2) < d 1 (x,y)+ε, which implies ) Π x (d 1 (x,y) sup f (t) L(B1,B 2) d 1 (x,y). Applying the inverse of Π x, we have d 1 (x,y) sup f (t) L(B1,B 2) Π 1 x (d 1 (x,y)), which implies that Ψ(d 1 (x,y)) sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2). To complete this proof we show that for any ε > 0, the value d 1 (x,y)+ε is not suitable for the triplet (f,x,r). Since y sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2) is a continuous application, than for all ξ > 0wecanfindζ > 0suchthatifd 1 (x,y) < ζ,then f (x) 1 L(B 1,B 2) sup f (t) L(B1,B 2) < ξ +1. In other words, if d 1 (x,y) < ζ then d 1 (x,y) sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2) f < d 1 (x,y)ξ +d 1 (x,y) < ζξ +d 1 (x,y). (x) L(B1,B 2)

15 A MEAN VALUE THEOREM FOR METRIC SPACES 15 Let M > 1 and set ξ = ε/m. Without loss of generality, we assume that ζ < 1. Hence, if d 1 (x,y) < ζ, then Choose λ y such that d 1 (x,y) sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2) f (x) L(B1,B 2) < d 1 (x,y)+ ε M. d 1 (x,y) sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2) f < λ y < d 1 (x,y)+ ε (x) L(B1,B 2) M. Since there is a direction v B 1 such that f (x) v B2 = f (x) L(B1,B 2), define h = λv and ỹ = x+h. Then d 1 (x,ỹ) = λ y < d 1 (x,y)+ ε M < d 1(x,y)+ε. On the other hand, f (x) h B2 = λ f (x) L(B1,B 2) > d 1 (x,y) sup f (t) L(B1,B 2). Choosing a suitable M and a sufficiently small value R > 0, if y B M1 (x,r) then f(x +h) f(x) f (x) h B2 is small enough, so f(ỹ) f(x) B2 d 1 (x,y) sup BM1 [x,d 1(x,y)] f (t) L(B1,B 2) even if d 1 (x,ỹ) < d 1 (x,y) + ε. Therewith, d 1 (x,y) + ε is not suitable for the desired triplet if y B M1 (x,r), which concludes the proof. Corollary 30. Suppose that M 1 and M 2 are open sets contained in Banach spaces B 1 and B 2, respectively, f : M 1 M 2 is continuously differentiable and f is enclosed at x. If there is a direction v B 1 such that f (x) v B1 = f (x) L(B1,B 2) 0, then, there exists R > 0 such that ) for all y B M1 (x,r). Π x (d 1 (x,y) sup f (t) L(B1,B 2) = d 1 (x,y) Corollary 31. Suppose that M 1 and M 2 are open sets contained in Banach spaces B 1 and B 2, respectively, f : M 1 M 2 is continuously differentiable and f is enclosed at x. If there is a direction v B 1 such that f (x) v B1 = f (x) L(B1,B 2) 0, then there exists R > 0 and a function Σ : B M1 (x,r) R + defined as that fulfills Σ(y) = d 1 (x,y) sup f (x) L(B1,B 2) (i) If y and ỹ are equidistant points from x, then Σ(y) = Σ(ỹ); (ii) If Σ(y) = Σ(ỹ), then y and ỹ are equidistant points from x. 7. The Mean Value Propriety The study of averages of analytic (or harmonic) functions comprises a huge literature (see for instance [6, 7, 10, 9]), and important results are proven by this theory. Our final intention in this paper is to relate such averages with our techniques, and prove some interesting properties when the averaging functional is evaluated on enclosed functions. Let (X,A,µ) be a measure space, f an integrable function, A a positive measure set and λ a scalar. We write M λ f A to indicate the shifted average of f over A. In other words, M λ f A = 1 µ(a) A f λdµ = 1 A A f λdµ.

16 16 P. M. CARVALHO NETO AND P. A. LIBONI FILHO Observe that if Mf A denotes the standard average over A, then we have that M λ f A = Mf A λ. Suppose that (X,d) is also a metric space and that A is the σ-algebra generated by the Borelian sets. It is a recurring task to determinate the maximum ball in which f stays bellow its average on certain fixed set. Let us start with some preliminary results. Lemma 32. If B(x,r 0 ) is a ball in X centered at x with radius r 0 and f : X C is an enclosed function at the same point x, then M f(x) f B(x,r0) Π 1 x (r 0 ) for all r 0 in the domain of the Π 1 x function. Proof. Sincer 0 isinthedomainoftheπ 1 x function, wehavethatifd(x,t) < r 0 then f(x) f(t) < Π 1 x (r 0 ). Therefore, f(t) f(x)dµ(t) f(t) f(x) dµ(t) Π 1 x (r 0 )dµ(t). B(x,r 0) B(x,r 0) B(x,r 0) Therefore, we connected the shifted average of a function f with the inverse of its continuity function. If we work with the average itself, we can just write Mf B(x,r0) f(x) Π 1 x (r 0 ). It is now clear that Π 1 x actually works as an upper bound for the difference between the image of the function in its center point and its average. Now suppose that f : X C is a strongly enclosed function at some point x and take some fixed number r 0 > 0. Additionally, suppose that f(x) = 0. Under these conditions, determining the maximum ball B(x,r) in which f stays bellow its average in B(x,r 0 ) is equivalent to determine the maximum r > 0 such that if d(x,t) < r, then f(t) < Mf B(x,r0). If Mf B(x,r0) liesinthedomainofπ x, thenr = Π x ( MfB(x,r0) ) isthesolutiontotheproblem. Since f(x) = 0, we have that Mf B(x,r0) = M f(x) f B(x,r0) Π 1 x (r 0 ). Using monotonicity arguments, we now obtain that r = Π x ( MfB(x,r0) ) r 0. In other words, if the average is taken on a ball of radius r 0, then the maximum ball that keeps f under its average has radius less or equal to r 0. Precisely, we have proved the Theorem 33. Let f : X C be a enclosed function at some point x and take a number r 0 > 0. Additionally, suppose that f(x) = 0 and that f is an unbounded function. Under these circumstances, there exists an r = r(r 0,x,f) 0 such that (i) If z B(x,r), then f(z) < Mf B(x,r0) ; (ii) If f is strongly enclosed at x, then r is the maximum radius satisfying the previous statement; (iii) If r 0 Π x (0, ), then r r 0. In addition to that, if f L 1 (X), then r 0 r(r 0 ) is a continuous application. Proof. We are almost done. First, note that since f is an unbounded function, we have that E f (x) = (0, ). Also, using the Dominated Convergence Theorem, we have that if f L 1 (X), then r 0 Mf B(x,r0) is a continuous application.

17 A MEAN VALUE THEOREM FOR METRIC SPACES 17 We can also state the following version of the theorem, which is a trivial consequence of the previous one. Theorem 34 (The Mean-Value Property for Enclosed Functions). Let f : X C be a enclosed function at some point x and take a number r 0 > 0. Additionally, suppose that f is an unbounded function. Under these circumstances, there exists an r = r(r 0,x,f) 0 such that (i) If z B(x,r), then f(z) f(x) < Mf B(x,r0) f(x) ; (ii) If f is strongly enclosed at x, then r is the maximum radius satisfying the previous statement; (iii) If r 0 Π x (0, ), then r r 0. In addition to that, if f L 1 (X), then r 0 r(r 0 ) is a continuous application. Some remarks are in order. First of all, the last theorem can be seen as a version of the Mean Value Property. In harmonic functions, one can expect that the image of the center of the ball is given by the average value of the function in the interior of the ball. In this version, for enclosed functions, we have shown that the image of any point which lies in a certain set is strictly small than the average taken on some larger ball. Also note that we call it a version not a generalization because of the following: suppose that f is a harmonic function. Under the assumptions of the last theorem, we have that Mf B(x,r0) f(x) = 0 and r = Π x (0) = 0. This is a natural and expected phenomenon, since harmonic functions behave much more nicely with respect to averages than enclosed functions. To conclude, observe that the unboundedness of the function should not pose a threat to the usability of the last theorem. Let us consider the following two cases. First, suppose that we have Π x (0, ) = (0, ) and r 0 R, for some R > 0. In this case, if there exists an unbounded function F such that F B(x,R) = f, then we can obtain the same conclusions. On the other hand, if we do not have any control over Π x (0, ), but R is small enough, then we can still claim the same conclusions. Corollary 35. Let f : X C be a enclosed function at some point x and take a number r 0 > 0. Additionally, suppose that f(x) = 0 and that f is an unbounded function. Under these circumstances, there exists an r = r(r 0,x,f) 0 such that Mf B(x,r) M f B(x,r) Mf B(x,r0). Acknowledgement The authors would like to thank the anonymous referee for his/her time and valuable comments. Moreover, the first author would like to thank the Federal University of São Carlos (UFSCar) for the hospitality and support during a short term visit in São Carlos. References [1] F. H. Clarke and Yu. S. Ledyaev, Mean value inequalities, Proc. Amer. Math. Soc. 122, (1994). [2] F.H. Clarke and Yu.S. Ledyaev, Mean value inequalities in Hilbert space, Trans. Amer. Math. Soc. 334, (1994). [3] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Qualitative properties of trajectories of control systems: a survey, J. Dynam. Control Systems 1, 1 48 (1995). [4] R. Deville, A mean value theorem for nondifferentiable mappings in Banach spaces, Serdica Math. J. 21, (1995). [5] J. Dieudonné, Foundations of Modern Analysis (Academic Press, New York, 1960). [6] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals, II. Math. Z. 34, (1932). [7] T. Kiso, Y. Mizuta and T. Shimomura, A theorem of Hardy-Littlewood for harmonic functions satisfying Hölder s condition, J. Math. Soc. Japan 47, (1995).

18 18 P. M. CARVALHO NETO AND P. A. LIBONI FILHO [8] A. S. Lewis and D. Ralph, A nonlinear duality result equivalent to the Clarke-Ledyaev mean value inequality, Nonlinear Anal. 26, (1996). [9] Y. Mizuta, A. Nekvinda, and T. Shimomura, Hardy averaging operator on generalized Banach function spaces and duality, Z. Anal. Anwend. 32, (2013). [10] A. Nekvinda and L. Pick, Optimal estimates for the Hardy averaging operator, Math. Nachr. 283, (2010). [11] D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91, (1990). [12] M. L. Radulescu and F. H. Clarke, The multidirectional mean value theorem in Banach spaces, Canad. Math. Bull. 40, (1197). [13] M. Turinici, Mean value theorems on abstract metric spaces, Math. Nachr. 115, (1984). [14] Q. Zhu, Clarke-Ledyaev mean value inequalities in smooth Banach spaces, Nonlinear Anal. 32, (1998). (P. M. Carvalho Neto) Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda 651, Campinas, Brazil. address: (P. A. Liboni Filho) Departamento de Matemática, Universidade Federal de São Carlos, Rod. Washington Luís, Km 235, São Carlos, Brazil. address:

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