aa r X i v : . [ m a t h . F A ] N ov ON THE ALMOST EVERYWHERE CONTINUITY
JO¨EL BLOT
Abstract.
The aim of this paper is to provide characterizations of the Lebesgue-almost everywhere continuity of a function f : [ a, b ] → R . These characteri-zations permit to obtain necessary and sufficient conditions for the Riemannintegrability of f . Key Words: continuity, function of one real variable, Riemann integrability.M.S.C. 2010: 26A15, 26A42. 1.
Introduction
The main aim of this paper is to establish the following theorem.
Theorem 1.1.
Let a > b be two real numbers, and f : [ a, b ] → R be a function.We assume that f admits a finite right-hand limit at each point of [ a, b ) excepton a Lebesgue-negligible set (respectively on a at most countable set). Then f iscontinuous at each point of [ a, b ] except on a Lebesgue-negligible set (respectively ona at most countable set). The origine of this work is a paper of Daniel
Saada [4] which states that a realfunction defined on a real segment which right-hand continuous possesses at mostan at most countable subset of discontinuity points. Saada attributes the proof ofthis result to Alain
R´emondi`ere . Studying this result and its proof, we see that itcontains a central argument that we have described in our Lemma 4.1 and Lemma4.2, and we use this argument to obtain other results. And so the present work isa continuation of the work of R´emondi`ere and Saada.In Section 2 we precise our notation and we give comments on them. In Section3, we establish lemmas which are useful for the proof of Theorem 1.1. In Section4, we provide results on the left-hand continuity and on the right-hand continuity.In Section 5 we give the proof of Theorem 1.1. In Section 6 we establish corollariesof Theorem 1.1. 2.
Notation
We use the left-hand oscillation of f at x ∈ ( a, b ] defined by ω L ( x ) := lim h → ( sup y ∈ [ x − h,x ] f ( y )) − lim h → ( inf y ∈ [ x − h,x ] f ( y ))and also the righ-hand oscillation of f at x ∈ [ a, b ) defined by ω R ( x ) := lim h → ( sup y ∈ [ x,x + h ] f ( y )) − lim h → ( inf y ∈ [ x,x + h ] f ( y )) . Date : November 4, 2014.
Note that we have sup y ∈ [ x − h,x ] f ( y ) ≥ f ( x ) and consequentlylim h → ( sup y ∈ [ x − h,x ] f ( y )) ≥ f ( x ) > −∞ . Also note that we have inf y ∈ [ x − h,x ] f ( y ) ≤ f ( x ) and consequentlylim h → ( inf y ∈ [ x − h,x ] f ( y )) ≤ f ( x ) < + ∞ . And so, ω L ( x ) is a sum of two elements of ( −∞ , + ∞ ] and therefore it is well-definedin ( −∞ , + ∞ ]; more precisely its belongs to [0 , + ∞ ]. For similar reasons, ω R ( x ) iswell-defined in [0 , + ∞ ].We use the following notation when g : [ a, b ] → ( −∞ , + ∞ ] and r ∈ R : { g = 0 } := { x ∈ [ a, b ] : g ( x ) = 0 } , { g > r } := { x ∈ [ a, b ] : g ( x ) > r } , { g ≤ r } := { x ∈ [ a, b ] : g ( x ) ≤ r } .A subset N ⊂ [ a, b ] is called Lebesgue-negligible when there exists B , a boreliansubset of [ a, b ], such that N ⊂ B and µ ( B ) = 0 where µ denotes the Lebesguemeasure of R . Such a vocabulary is used for instance in [1]. Remark 2.1.
The following equivalence hold. (A)
When x ∈ ( a, b ] , ω L ( x ) = 0 if and only if f is left-hand continuous at x . (B) When x ∈ [ a, b ) , ω R ( x ) = 0 if and only if f is right-hand continuous at x . (C) When x ∈ ( a, b ) , ( ω L ( x ) = 0 and ω R ( x ) = 0 ) if and only if f is continuousat x . These equivalences are easy to prove. One important fact is that x belongs tothe neighborhoods [ x − h, x ] and [ x, x + h ]. Remark 2.2.
When we will speak of the left-hand limit (respectively of the right-hand limit) of the function f at x , we speak of the limit of f ( y ) when y → x, y > x (respectively y → x, y < x ); the point x is not included into his ”‘neighborhoods”’.The situation is different in the definition of the oscillations ω L and ω R . We denote f ( x − ) := lim y → x,y
Lemma 3.1.
Let f : [ a, b ] → R be a function, and z ∈ [ a, b ) . We assume that f admits a finite right-hand limit at z . Then we have : ∀ ǫ > , ∃ λ ( z, ǫ ) > , ∀ x ∈ ( z, z + λ ( z, ǫ )] , ω L ( x ) ≤ ǫ. Proof.
We arbitrarily fix ǫ >
0. Using the assumption, there exists d z ∈ R suchthat ∃ η ( z, ǫ ) > , ∀ x ∈ [ a, b ] , z < x ≤ z + η ( z, ǫ ) = ⇒ | f ( x ) − d z | ≤ ǫ. (3.1)When x ∈ ( z, z + η ( z, ǫ )] and when y ∈ ( z, x ], we have y ∈ ( z, z + η ( z, ǫ )], andusing (3.1) we obtain | f ( x ) − d z | ≤ ǫ and | f ( x ) − d z | ≤ ǫ that implies | f ( x ) − f ( y ) | ≤ | f ( x ) − d z | + | f ( y ) − d z | ≤ ǫ ǫ ⇒ f ( x ) − ǫ ≤ f ( y ) ≤ f ( x ) + ǫ . ONTINUITY 3
Then, for all h ∈ (0 , x − z ], we obtain f ( x ) − ǫ ≤ inf y ∈ [ x − h,x ] f ( y ) ≤ sup y ∈ [ x − h,x ] f ( y )) ≤ f ( x ) + ǫ ⇒ f ( x ) − ǫ ≤ lim h → ( inf y ∈ [ x − h,x ] f ( y )) ≤ lim h → ( sup y ∈ [ x − h,x ] f ( y )) ≤ f ( x ) + ǫ ⇒ ≤ ω L ( x ) ≤ f ( x ) + ǫ − ( f ( x ) − ǫ ǫ. And so it suffices to take λ ( z, ǫ ) := η ( z, ǫ ). (cid:3) Using similar arguments we can prove the following result.
Lemma 3.2.
Let f : [ a, b ] → R be a function, and z ∈ ( a, b ] . We assume that f admits a finite left-hand limit at z . Then we have : ∀ ǫ > , ∃ ν ( z, ǫ ) > , ∀ x ∈ [ z − ν ( z, ǫ ) , z ) , ω R ( x ) ≤ ǫ. Lemma 3.3.
Let I be a nonempty set, and ( S i ) i ∈ I be a family of subintervals of [ a, b ] such that S i ∩ S j = ∅ when i = j , and such µ ( S i ) > for all i ∈ I , where µ denotes the Lebesgue measure of R . Then I is at most countable.Proof. Since a positive measure is additive, for all finie subset J ⊂ I , we have µ ( ∪ j ∈ J S j ) = P j ∈ J µ ( S j ). Since a positive measure is monotonic, ∪ j ∈ J S j ⊂ [ a, b ]implies µ ( ∪ j ∈ J S j ) ≤ µ ([ a, b ]) = b − a , and so we have P j ∈ J µ ( S j ) ≤ b − a < + ∞ forall finite subset J of I . Therefore the family of non negative real numbers ( µ ( S i )) i ∈ I is summable in [0 , + ∞ ), and consequently the set { i ∈ I : µ ( S i ) = 0 } is at mostcountable (Corrolary 9-9, p. 220 in [2]). Since µ ( S i ) > i ∈ I , we obtainthat I is at mot countable. (cid:3) Remark 3.4.
We can also prove Lemma 3.3 by building a function ϕ : I → Q inthe following way: since Q is dense into R , for each i ∈ I , there exists ϕ ( i ) ∈ Q ∩ S i .Since I i ∩ I j = ∅ when i = j , we have ϕ ( i ) = ϕ ( j ) when i = j . And so ϕ is injective.Since Q is countable, ϕ ( I ) ⊂ Q is at most countable, and using an abridgement of ϕ , we build a bijection between ϕ ( I ) and I . Limits on one side, continuities on the other side
The following results establish that the existence of left-hand (respectively right-hand) limits implies the right-hand (respectively left-hand) continuity.
Lemma 4.1. let f : [ a, b ] → R be a function, and N be a Lebesgue-negligible(respectively at most countable) subset of [ a, b ] . We assume that f admits a finiteright-hand limit at each x ∈ [ a, b ) \ N .Then the set of the points of [ a, b ] where f is not left-hand continuous is Lebesgue-negligible (respectively at most countable).Proof. We arbitrarily fix ǫ >
0. Using Lemma 3.1, denoting λ z := λ ( z, ǫ ), we obtainthe following assertion. ∀ z ∈ { ω L > ǫ } ∩ ([ a, b ] \ N ) , ∃ λ z > , ( z, z + λ z ] ⊂ { ω L ≤ ǫ } . (4.1) BLOT
Let z , z ∈ { ω L > ǫ } ∩ ([ a, b ] \ N ), z = z . We can assume that z < z . After(4.1), we cannot have z into ( z , z + λ z ], therefore we have z > z + λ z , andwe have proven: ∀ z , z ∈ { ω L > ǫ } ∩ ([ a, b ] \ N ) , z = z = ⇒ ( z , z + λ z ] ∩ ( z , z + λ z ] = ∅ . We have also µ (( z, z + λ z ]) = λ z >
0. Then using Lemma 3.3, we can assert that ∀ ǫ > , { ω L > ǫ } ∩ ([ a, b ] \ N ) is at most countable . (4.2)Note that { ω L > } = ω − L ((0 , + ∞ ]) = ω − L ( S n ∈ N ∗ ( n , + ∞ ])= S n ∈ N ∗ ω − L (( n , + ∞ ]) = S n ∈ N ∗ { ω L > n } = ⇒{ ω L > } ∩ ([ a, b ] \ N ) = [ n ∈ N ∗ ( { ω L > n } ∩ ([ a, b ] \ N )) . Using (4.2), since a countable union of at most countable subsets is at most count-able, we ontain the following assertion. { ω L > } ∩ ([ a, b ] \ N ) is at most countable . (4.3)Note that { ω L > } = ( { ω L > } ∩ ([ a, b ] \ N )) ∪ ( { ω L > } ∩ N ).Since ( { ω L > } ∩ N ) ⊂ N and since N is Lebesgue-negligible (respectively at mostcountable), ( { ω L > } ∩ N ) is Lebesgue-negligible (respectively at most countable).Recall that an at most countable subset of R is Lebesgue-negligible. And so when N is Lebesgue-negligible, { ω L > } is Lebesgue-negligible as a union of two Lebesgue-negligible susbets, and when N is at most countable, { ω L > } is at most countableas a union of two at most countable subsets. Using (A) of Remark 2.1, the lemmais proven. (cid:3) Proceegings as in the proof of Lemma 4.1, we obtain the following result.
Lemma 4.2. let f : [ a, b ] → R be a function, and M be a Lebesgue-negligible(respectively at most countable) subset of [ a, b ] . We assume that f admits a finiteleft-hand limit at each x ∈ ( a, b ] \ M .Then the set of the points of [ a, b ] where f is not right-hand continuous is Lebesgue-negligible (respectively at most countable). Proof of Theorem 1.1
Using Lemma 4.1 and (A) of Remark 2.1, { ω L > } is Lebesgue-negligible (re-spectively at most countable) since { ω L > } is exactly the set of the points of ( a, b ]where f is not left-hand continuous.Now, setting M = { ω L > } , for all x ∈ [ a, b ] \ M , f ( x − ) = f ( x ) ∈ R , and theassumption of Lemma 4.2 is fulfilled. Consequently we obtain that { ω R > } isLebesgue-negligible (respectively at most countable) after (B) of Remark 2.1.Note that { ω L = 0 } ∩ { ω R = 0 } is exactly the set of the points of ( a, b ) where f is continuous. We have[ a, b ] \ ( { ω L = 0 }∩{ ω R = 0 } ) = [ a, b ] ∩ ( { ω L > }∪{ ω R > } ) = { ω L > }∪{ ω R > } . This set is Lebesgue-negligible (respectively at most countable) as a union oftwo Lebesgue-negligible (respectively at most countable) sets. Note that { a, b } isLebesgue-negligible (respectively at most countable) and so set of the discontinuitypoints of f is Lebesgue-negligible (respectively at most countable). ONTINUITY 5 Consequences
A first consequence of Theorem 1.1 is the following result.
Theorem 6.1.
Let a > b be two real numbers, and f : [ a, b ] → R be a function.Then the following assertions are equivalent. ( α ) The set of the discontinuity points of f is Lebesgue-negligible (respectivelyat most countable). ( β ) The set of the left-hand discontinuity points of f is Lebesgue-negligible (re-spectively at most countable). ( γ ) The set of the right-hand discontinuity points of f is Lebesgue-negligible(respectively at most countable). ( δ ) The set of the points where f does not admit a finite left-hand limit isLebesgue-negligible (respectively at most countable). ( ǫ ) The set of the points where f does not admit a finite right-hand limit isLebesgue-negligible (respectively at most countable).Proof. The implications ( α ) = ⇒ ( β ) = ⇒ ( δ ) are easy, and ( δ ) = ⇒ ( α ) is Theorem1.1. The implications ( α ) = ⇒ ( γ ) = ⇒ ( ǫ ) are easy. we can do a proof which issimilar to this one of Theorem 1.1 to prove ( ǫ ) = ⇒ ( α ). (cid:3) About the Riemann-integrability we recall a famous theorem of Lebesgue, [3] p.29, [5] p. 20.
Theorem 6.2.
Let a > b be two real numbers, and let f : [ a, b ] → R be a boundedfunction. Then the following assertions are equivalent. (i) f is Riemann integrable on [ a, b ] . (ii) The set of the discontinuity points of f is Lebesgue-negligible. As a consequence of Theorem 6.1 and of the previous classical theorem of Lebesgue,we obtain the following result on the Riemann-integrability.
Theorem 6.3.
Let a > b be two real numbers, and let f : [ a, b ] → R be a boundedfunction. Then the following assertions are equivalent. ( a ) f is Riemann integrable on [ a, b ] . ( b ) The set of the points where f does not admit a finite left-hand limit isLebesgue-negligible. ( c ) The set of the points where f does not admit a finite right-hand limit isLebesgue-negligible. An easy consequence of this result is the following one.
Corollary 6.4.
Let a > b be two real numbers, and let f : [ a, b ] → R be a function.If f is right-hand continuous on [ a, b ] or left-hand continuous on [ a, b ] , then theset of the discontinuity points of f is at most countable, and consequently when inaddition f is assumed to be bounded, f is Riemann integrable on [ a, b ] . Acknowledgements.
I thanks my colleagues B. Nazaret, M. Bachir and J.-B.Baillon for interesting discussions on these topics.
BLOT
References [1] G. Choquet,
Lectures on analysis; Volume 1: integration and topological vectors spaces , W.A.Benjamin, Inc., Reading, Massachussets, 1969.[2] G. Choquet,
Topologie , second revised edition, Masson et Cie, Paris, 1973.[3] H. Lebesgue,
Le¸cons sur l’int´egration et la recherche des fonctions primitives , second edition,Gauthier-Villars, Paris, 1928; reprinted by Chelsea Publishing Company, Bronx, New York,1973[4] D. Saada,
Fonctions continues presque-partout
Integral, measure and derivative: a unified approach , Prentice-Hall, Inc., Upper Saddle River, NJ, 1966; reprinted by Dover Publications, Inc., New York,1977.
Jo¨el Blot: Laboratoire SAMM EA 4543,Universit´e Paris 1 Panth´eon-Sorbonne, centre P.M.F.,90 rue de Tolbiac, 75634 Paris cedex 13, France.
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