Forcing Differentiable Functions

F orcing and Differen tiable F unctions ∗ Kenneth Kunen † No v ember 1, 2018 Abstract W e consider co v ering ℵ 1 × ℵ 1 rectangles by co unta bly many smooth curv es, and differen tiable isomorphisms b et w een ℵ 1 -dense sets of reals. 1 In tro duct i on In this paper, w e consider t w o differen t iss ues, b oth related to the question of ob- taining differen tiable real–v alued functions where classical results only pro duced functions or contin u ous functions. Regarding the first issue, the text of Sierpi ´ nski [11] sho ws tha t CH is equiv alen t to his Prop osition P 2 , whic h is the statemen t that the pla ne “est une somme d’une infinit ´ e d ´ enom brable de courb es”. Here, a “curv e” is just the graph of a function or an inv erse function, so P 2 sa ys only that R 2 = S i ∈ ω ( f i ∪ f − 1 i ), where eac h f i is (the graph of ) a function f r o m R to R , with no assumption of con tin uit y . The pro of actually sho ws, in ZFC, that for ev ery E ∈ [ R ] ℵ 1 , there are f i : R → R with E 2 ⊆ S i ∈ ω ( f i ∪ f − 1 i ), and that this is false for all E of size greater than ℵ 1 . Usually in geometry and ana lysis, “curv e” do es imply con tin uity , so it is nat- ural to ask whether the f i can a ll b e con tin uo us, or ev en C ∞ : Definition 1.1 F or n ∈ ω ∪ {∞} , c al l E ⊆ R n –small iff ther e ar e C n functions f i : R → R such that E 2 ⊆ S i ∈ ω ( f i ∪ f − 1 i ) . Her e, C 0 just me ans “c ontinuous”, and C ∞ me ans C n for al l n ∈ ω . ∗ 2010 Mathematics Sub ject Classifica tion: Primary 03 E35, 03E5 0. K ey W or ds and P hr ases: PF A, MA, ℵ 1 -dense, curve. † Univ ersity of Wiscons in, Madiso n, WI 5 3706, U.S.A., kunen@math.wisc.edu 1 1 INTR ODUCTION 2 Coun table sets a re trivially ∞ –small, a nd b y Sierpi ´ nski, | E | ≤ ℵ 1 for ev ery 0–small set E , so w e are only in terested in sets of siz e ℵ 1 . Ev ery 0–small set is of first category and measure 0 (and perfectly meager and univ ersally n ull). Just in ZF C, we shall pro v e the following in Section 2: Theorem 1.2 Ther e is an E ∈ [ R ] ℵ 1 which is ∞ –smal l. The existence of a 0– small set is due to Kubi ´ s a nd V ejnar [8]. But no w w e can ask whe ther every E ∈ [ R ] ℵ 1 is n –small f or some n . Ev en when n = 0, this w ould imply that ev ery suc h E is of first category and measure 0 (a nd p erfectly meager and univ ersally n ull), whic h is a we ll-kno wn conseque nce of MA( ℵ 1 ). In fact, t he follo wing theorem fo llows easily from r esults already in the literatur e, as w e shall p oint out in Section 2: Theorem 1.3 1. MA( ℵ 1 ) imp lies that every set of s i z e ℵ 1 is 0 –smal l. 2. PF A implies that every set of si z e ℵ 1 is 1 – smal l. 3. MA( ℵ 1 ) do es not im ply that e v ery set of size ℵ 1 is 1 – smal l. 4. In ZF C , ther e is an E ∈ [ R ] ℵ 1 which is not 2 –smal l. W e r emark t hat Sierpi ´ nski’s use of “curv e” is unus ual in anot her wa y: Usually , w e w ould call a subset of R 2 a curv e iff it is a contin uous image of [0 , 1], a nd not necessarily the graph of a function; but with that usage, t he plane is a lw ays a coun table union o f curv es b y Pe ano [10]. Our second issue in v olves the isomorphism of ℵ 1 -dense subsets of R . Definition 1.4 E ⊆ R is ℵ 1 -dense iff | E ∩ ( x, y ) | = ℵ 1 whenever x, y ∈ R and x < y . F is the set of al l or der-p r eserving bije ctions fr om R onto R . By Baumgartner [3 , 4], PF A implies that whenev er D , E are ℵ 1 -dense, there is an f ∈ F suc h that f ( D ) = E . By [2, 1], this cannot b e prov ed from MA( ℵ 1 ) alone. Clearly , ev ery f ∈ F is con tin uous, but w e can ask whether w e can alw ays get our f to b e C n . F or n = 2, a ZF C coun ter-example is apparen t f rom Theorems 1.2 a nd 1.3, since w e ma y take D to b e 2–small a nd E to b e not 2–small, and also assume that D = D + Q = { D + q : q ∈ Q } and E = E + Q . Note that D is 2–small iff D + Q is 2–small, a nd the latter set is also ℵ 1 -dense. But in f a ct, ev en n = 1 is imp ossible, since the f o llo wing holds in ZF C, as we shall show in Section 3 : 1 INTR ODUCTION 3 Theorem 1.5 Ther e a r e ℵ 1 -dense D , E ⊂ R such that for al l f ∈ F and ℵ 1 - dense D ∗ ⊆ D and E ∗ ⊆ E with f ( D ∗ ) = E ∗ : If p < q and a = f ( p ) and b = f ( q ) then: 1. Either f is not uniformly Lipschitz on ( p, q ) or f − 1 is not uniformly Lip- schitz on ( a, b ) ; e quivalently, whenever 0 < Λ ∈ R , ther e ar e x 0 , x 1 ∈ ( p, q ) such that either | f ( x 1 ) − f ( x 0 ) | > Λ | x 1 − x 0 | or | x 1 − x 0 | > Λ | f ( x 1 ) − f ( x 0 ) | . 2. Either f ′ do es not exist at some d ∈ D ∗ ∩ ( p, q ) or ( f − 1 ) ′ do es not exist at some e ∈ E ∗ ∩ ( a, b ) . 3. If f ′ ( d ) exists for al l d ∈ D ∗ ∩ ( p, q ) , then f ′ ( d ) = 0 for al l but c ountably many d ∈ D ∗ ∩ ( p, q ) . In particular, f cannot b e in C 1 ( R ), since f ′ cannot b e 0 ev erywhere, so if f ′ w ere contin uous, there w ould b e an inte rv al on which f ′ > 0, con tradicting (3). On the o ther hand, f ′ can exist ev erywhere and b e 0 on a dense set if f ′ is not required to b e con tin uous: Theorem 1.6 Assume PF A , an d let D , E ⊂ R b e ℵ 1 -dense. Then ther e exist f ∈ F and D ∗ ⊆ D such that D ∗ is ℵ 1 -dense and f ( D ∗ ) = E and 1. F or al l x ∈ R , f ′ ( x ) exists and 0 ≤ f ′ ( x ) ≤ 2 . 2. f ′ ( d ) = 0 for al l d ∈ D ∗ . By (1), f satisfies a uniform Lipsc hitz condition with Lipsc hitz constan t 2. The “2 ” is an artif a ct of the pro of, a nd ma y b e replaced b y an arbitrarily small n um b er; if ε > 0, w e can get o ur f with f ′ ( x ) ≤ 2 so that f ( D ∗ ) = (1 /ε ) E ; then εf ′ ( x ) ≤ 2 ε and εf ( D ∗ ) = E . In (2), the f ′ ( d ) = 0 is to b e exp ected, in view of Theorem 1 .5(3). W e do not know whether we can mak e D ∗ equal D . The pro o f of Theorem 1 .6 in Sections 4 and 5 a ctually show s that one can force the result to hold in an a ppro priate ccc extension of any mo del of ZFC + 2 ℵ 0 = ℵ 1 + 2 ℵ 1 = ℵ 2 . Then the result follo ws from PF A using t he same fo rcing plus the “collapsing the contin uum” tric k. W e remark that Theorem 1.6 con tradicts Prop o sition 9.4 in the pap er [1] of Abraham, Rubin, and Shelah, whic h pro duces a ZF C example of ℵ 1 -dense D , E ⊂ R suc h that ev ery f ∈ F with f ∩ ( D × E ) uncoun table fails to b e differen tiable at uncountably many elemen t s of D . Their “pro of ” uses ideas similar to our pr o of of Theorem 1.5, but insufficien t details are giv en to b e able to lo cate a sp ecific error. 2 ON SMALLNESS 4 2 On Smallness W e first p oint out that Theorem 1.3 follow s easily from kno wn results: Pro of of T heorem 1.3. F or (1 ) , fix E ∈ [ R ] ℵ 1 . By Sierpi ´ nski, E 2 ⊆ S i ∈ ω ( f i ∪ f − 1 i ), where eac h f i is the graph of a function and | f i | = ℵ 1 . Then , assuming MA( ℵ 1 ), a standard forcing sho ws t ha t for eac h i , there are Cantor sets P i,n for n ∈ ω with eac h P i,n the gr a ph of a function a nd f i ⊆ S n P i,n . Now eac h P i,n extends to a function g i,n ∈ C ( R , R ), so that E 2 ⊆ S i,n ( g i,n ∪ g − 1 i,n ). F or (2), use the fact from [5] that under PF A, ev ery A ∈ [ R 2 ] ℵ 1 is a subset of a countable union of C 1 arcs. No w apply this with A = E × E , and note that ev ery C 1 arc is contained in a finite union of (graphs o f ) C 1 functions and in verse functions. (4) also follo ws from [5], whic h sho ws in ZFC that there is an A ∈ [ R 2 ] ℵ 1 whic h is not a subset o f a coun table union of C 2 arcs. So , c ho ose E suc h that A ⊆ E × E . Lik ewise , (3) follo ws from [9], whic h sho ws that it is consisten t with MA( ℵ 1 ) to ha v e an A ∈ [ R 2 ] ℵ 1 whic h is a w eakly Luzin set; and suc h a set is not a subset of a countable union o f C 1 arcs. K Next, to prov e Theorem 1.2, w e first state an abstract v ersion of the argumen t in v o lv ed: Lemma 2.1 Supp ose that T is an unc ountable set with functions f i on T for i ∈ ω such that for al l c ountable Q ⊂ T , ther e is an x ∈ T such that Q ⊆ { f i ( x ) : i ∈ ω } . The n ther e is an E ⊆ T of siz e ℵ 1 such that E × E ⊆ ∆ ∪ S i ( f i ∪ f − 1 i ) , wher e ∆ is the identity function . Pro of. Note, by considering sup ersets of Q , that there m ust b e uncountably man y suc h x . No w, let E = { e α : α < ω 1 } where e α is c hosen recursiv ely so that e α / ∈ { e ξ : ξ < α } ⊆ { f i ( e α ) : i ∈ ω } . K T o illustrate the idea of our ar g umen t, w e first pro duce an E ∈ [ R ] ℵ 1 whic h is 0–small, in whic h case T can b e a ny Cantor set. Lemma 2.2 Ther e ar e f i ∈ C (2 ω , 2 ω ) for i < ω such that for al l c ountable n on- empty Q ⊆ 2 ω , ther e is an x ∈ 2 ω such that Q = { f i ( x ) : i < ω } . Pro of. Let ϕ map ω × ω 1-1 in to ω , and let ( f i ( x ))( j ) = x ( ϕ ( i, j )). No w, let Q = { y i : i ∈ ω } . Since ϕ is 1- 1 , w e ma y choose x ∈ 2 ω suc h that x ( ϕ ( i, j )) = y i ( j ) for all i, j ; then f i ( x ) = y i . K 2 ON SMALLNESS 5 So, if T ⊆ R is a Cantor set, then T ∼ = 2 ω , a nd the existence of an E ∈ [ T ] ℵ 1 whic h is 0– small follows from Lemmas 2.1 and 2.2 , and the observ ation that ev ery function in C ( T , T ) extends to a function in C ( R , R ). No w, if w e w an t our f unctions to b e smo oth, as required by Theorem 1.2 , w e m ust b e a bit more careful. The f i will b e defined on the standard middle-third Can tor set H , but t hey will only satisfy L emma 2.1 on a thin subset T ⊂ H . T o simplify notatio n, H will b e a subset of [0 , 3] rather than [0 , 1]. F or x ∈ [0 , 3], x ∈ H iff x has only 0s and 2 s in its ternary expansion, so that x = P n ∈ ω x ( n )3 − n , where each x ( n ) ∈ { 0 , 2 } , and we write x in ternary as x (0) .x (1) x (2) x (3) x (4) · · · . If x, y ∈ H with x 6 = y , let δ ( x, y ) b e the least n suc h that x ( n ) 6 = y ( n ), and note tha t 3 − n ≤ | x − y | ≤ 3 − n +1 . Fix an y Γ : ω → ω suc h that Γ(0) = 0, Γ is strictly increasing, and Γ( k + 1) ≥ (Γ( k )) 2 for eac h k . The minim um suc h Γ is the seq uence 0 , 1 , 2 , 4 , 16 , 256 , . . . , but an y other suc h Γ will do. W e view x in H as co ding an ω –sequence of blo cks , where the k th blo c k is a sequence of length Γ( k + 1) − Γ ( k ). Note that Γ( k + 2) − Γ( k + 1) ≥ Γ( k + 1) − Γ( k ) for each k , so the blo ck s get longer as k ր . More for mally , fo r x ∈ H and k ≥ 0, w e define B x k : ω → { 0 , 2 } so that B x k ( j ) = x (Γ( k )+ j ) when j < Γ( k +1) − Γ( k ) and B x k ( j ) = 0 f o r j ≥ Γ( k +1) − Γ( k ). Note t ha t x is determined by h B x k : k ∈ ω i . Let B x k ( j ) = 0 when k < 0. No w, we wish x ∈ H to enco de a sequence o f ω elemen t of H , h f i ( x ) : i ∈ ω i . W e do t his using a bijection ϕ f r o m ω × ω on to ω . W e assume that max ( i, j ) < max( i ′ , j ′ ) → ϕ ( i, j ) < ϕ ( i ′ , j ′ ) fo r all i, j, i ′ , j ′ , whic h implies that max( i, j ) 2 ≤ ϕ ( i, j ) < (max( i, j ) + 1) 2 . In the “standard” enco ding, as in the pro o f of Lemma 2.2, an x ∈ { 0 , 2 } ω can enco de ω elemen ts o f { 0 , 2 } ω , where the i th elemen t is j 7→ x ( ϕ ( i, j )). But here, for x ∈ K , w e apply this separately to eac h of the ω blo cks of x , and w e shift righ t tw o places to ensure that the functions are smo oth. Define f i : H → H so that for x ∈ H , f i ( x ) is the z ∈ H such that B z k ( j ) = B x k − 2 ( ϕ ( i, j )) for a ll j ; so B z k ( j ) = 0 when k < 2. There is suc h a z b ecause j ≥ Γ( k + 1) − Γ( k ) ⇒ ϕ ( i, j ) ≥ j ≥ Γ( k − 1) − Γ( k − 2) ⇒ B z k ( j ) = 0 . Let S = { 0 , 2 } <ω . F or i ∈ ω and s ∈ S , define f s i : H → H so that for x ∈ H , f s i ( x ) is the z ∈ H suc h that z ( n ) is s ( n ) f o r n < lh( s ) and f i ( x )( n ) fo r n ≥ lh( s ). Note t ha t most elemen ts of H are not in S { f s i ( H ) : i ∈ ω & s ∈ S } , but the T of Lemma 2.1 will b e a pro p er subset of H . First, w e v erify that w e get C ∞ functions. F ollow ing [5], call f : H → H flat iff for a ll q ∈ ω , there is a b ound M q suc h that f o r all u, t ∈ H , | f ( u ) − f ( t ) | ≤ M q | u − t | q . By Lemma 6.4 of [5], this implies t ha t f can b e extended to a C ∞ function defined on all of R , all of whose deriv ativ es v anish on H . 2 ON SMALLNESS 6 Lemma 2.3 Each f s i is flat. Pro of. Fix x, y in H with x 6 = y . Let n = δ ( x, y ). Fix k ∈ ω so that Γ( k ) ≤ n < Γ( k + 1). Then Γ( k + 2) ≤ δ ( f s i ( x ) , f s i ( y )). No w | x − y | ≥ 3 − n ≥ 3 − Γ( k + 1) , and | f s i ( x ) − f s i ( y ) | ≤ 3 − Γ( k + 2)+1 , so | f s i ( x ) − f s i ( y ) | / | x − y | q ≤ 3 − Γ( k + 2)+1+ q Γ( k +1) ≤ 3 − (Γ( k +1)) 2 +1+ q Γ( k +1) , whic h is b ounded, and in fact go es to 0 as k ր ∞ . K No w, w e define T ⊂ H : F o r x ∈ H and k ∈ ω , let ℓ x k b e the least ℓ ∈ ω suc h that ∀ j ≥ ℓ [ B x k ( j ) = 0]. So , ℓ x k ≤ Γ( k + 1) − Γ( k ). Call ψ : ω → ω tiny iff lim k →∞ ( ψ ( k ) n ) /k = 0 for all n ∈ ω . Note that t ininess is preserv ed by p o w ers a nd shifts. That is, if ψ is tin y , then so is k 7→ ψ ( k ) r and k 7→ r + ψ ( k + r ) for eac h r > 0. Pro of of Theorem 1.2. Let T b e the set of x ∈ H suc h that k 7→ ℓ x k is tin y . Then T is an uncoun table Bo rel set, and we are done by Lemma 2.4: K Lemma 2.4 If y i ∈ T for i ∈ ω , then ther e is an x ∈ T an d s i ∈ S for i ∈ ω such that f s i i ( x ) = y i for al l i . Pro of. F ix any ψ : ω → ω such that ψ ( k ) ≤ Γ( k + 1) − Γ( k ) for all k . Then w e can define x ∈ H so that B x k ( ϕ ( i, j )) = B y i k +2 ( j ) whenev er ϕ ( i, j ) < ψ ( k ); let B k x ( m ) = 0 for m ≥ ψ ( k ). Then x ∈ T provide d that ψ is tiny . F or each i , the function k 7→ ( i + ℓ y i k +2 ) 2 is t iny . No w, fix a tiny ψ suc h that ψ ( k ) ≤ Γ( k + 1) − Γ( k ) for all k ∈ ω and ψ ≥ ∗ ( k 7→ ( i + ℓ y i k +2 ) 2 ) for eac h i ; this is p ossible b y a standard diago nal argumen t. No w fix i . Then fix r ∈ ω suc h that ψ ( k ) ≥ ( i + ℓ y i k +2 ) 2 for all k ≥ r . Let s i = y i ↾ Γ( r + 2 ). Let z = f s i i ( x ). W e shall sho w that z = y i . So, fix n ∈ ω , and we sho w that z ( n ) = y i ( n ). This is ob vious if n < Γ( r + 2), so assume that n ≥ Γ( r + 2). Then fix k ≥ r + 2 and j < Γ( k + 1) − Γ ( k ) with n = Γ( k ) + j . W e m ust show that B z k ( j ) = B y i k ( j ). By definition of f s i i , B z k ( j ) = B x k − 2 ( ϕ ( i, j )), whereas w e only know t ha t B y i k ( j ) = B x k − 2 ( ϕ ( i, j )) when ϕ ( i, j ) < ψ ( k − 2). So, assume that ϕ ( i, j ) ≥ ψ ( k − 2 ); w e show that B z k ( j ) = 0 and B y i k ( j ) = 0. No w B y i k ( j ) = 0 b ecause otherwise j < ℓ y i k , and then ϕ ( i, j ) ≤ ( i + j ) 2 < ( i + ℓ y i k ) 2 ≤ ψ ( k − 2), a con tradiction. Also, B z k ( j ) = B x k − 2 ( ϕ ( i, j )) = 0 b y the definition of x , since ϕ ( i, j ) ≥ ψ ( k − 2). K 3 NON-ISOMORPHISMS 7 3 Non-Isomorp h isms Here w e prov e Theorem 1.5. First, Lemma 3.1 Ther e ar e Cantor sets H, K ⊂ R such that ∀ ε > 0 ∃ δ > 0 ∀ x 0 , x 1 ∈ H ∀ y 0 , y 1 ∈ K  0 < | x 1 − x 0 | < δ ∧ 0 < | y 1 − y 0 | < δ − → ( y 1 − y 0 ) / ( x 1 − x 0 ) ∈ ( − ε, ε ) ∪ (1 /ε, ∞ ) ∪ ( −∞ , − 1 /ε )  . Pro of. W e o btain H , K b y the usual trees of closed interv als: 1. H = T n ∈ ω S { I σ : σ ∈ n 2 } and K = T n ∈ ω S { J τ : τ ∈ n 2 } . 2. I σ = [ a σ , b σ ] a nd J τ = [ c τ , d τ ]. 3. a σ = a σ ⌢ 0 < b σ ⌢ 0 < a σ ⌢ 1 < b σ ⌢ 1 = b σ . 4. c τ = c τ ⌢ 0 < d τ ⌢ 0 < c τ ⌢ 1 < d τ ⌢ 1 = d τ 5. Whenev er lh( σ ) = lh( τ ) = n : b σ − a σ = p n and b τ − a τ = q n . Informally , assume that lh( σ ) = lh( τ ) = n . Then I σ × J τ is a b ox of dimensions p n × q n . It will b e v ery long and skinn y ( p n ≫ q n ). Inside this b o x will b e four little b oxes , of dimensions p n +1 × q n +1 , situated at the corners of the p n × q n b o x. These litt le ones are muc h smaller; that is, p n ≫ q n ≫ p n +1 ≫ q n +1 . Now supp ose that the t w o p oints ( x 0 , y 0 ) and ( x 1 , y 1 ) b oth lie in I σ × J τ , but lie in differen t smaller b ox es I σ ⌢ µ × J τ ⌢ ν . So, there are  4 2  = 6 p o ssibilities . F or t w o of them, b et w een I σ ⌢ µ × J τ ⌢ 0 and I σ ⌢ µ × J τ ⌢ 1 ( µ ∈ { 0 , 1 } ), the slop e | ∆ y / ∆ x | is v ery large. F or the o ther four, b et wee n I σ ⌢ 0 × J τ ⌢ ν and I σ ⌢ 1 × J τ ⌢ ν ( ν ∈ { 0 , 1 } ), or b et w een I σ ⌢ 0 × J τ ⌢ 0 and I σ ⌢ 1 × J τ ⌢ 1 or b et w een I σ ⌢ 0 × J τ ⌢ 1 and I σ ⌢ 1 × J τ ⌢ 0 , | ∆ y / ∆ x | is very small. More f o rmally , assume t ha t p 0 > q 0 > p 1 > q 1 > · · · and q n /p n → 0 and p n +1 /q n → 0 as n → ∞ . Fix ( x 0 , y 0 ) and ( x 1 , y 1 ) in H × K , and then fix n suc h that for some σ , τ ∈ n 2, ( x 0 , y 0 ) , ( x 1 , y 1 ) ∈ I σ × J τ , but ( x 0 , y 0 ) , ( x 1 , y 1 ) are in t w o differen t smaller b oxes I σ ⌢ µ × J τ ⌢ ν . Note that this n → ∞ as δ → 0. In the t w o large slop e cases, | ∆ y / ∆ x | ≥ ( q n − 2 q n +1 ) /p n +1 → ∞ as n → ∞ , since q n /p n +1 → ∞ and q n +1 /p n +1 → 0. In the four small slop e cases, | ∆ y / ∆ x | ≤ q n / ( p n − 2 p n +1 ) → 0, since p n /q n → ∞ and p n +1 /q n → 0. K Pro of of Theorem 1.5. Fix H , K a s in Lemma 3.1, and then fix ˜ H ∈ [ H ] ℵ 1 and ˜ K ∈ [ K ] ℵ 1 . Let D = S { ˜ H + s : s ∈ Q } and E = S { ˜ K + t : t ∈ Q } . No w, fix f , D ∗ , E ∗ , p, q , a, b as in Theorem 1.5. Then the function f ∗ := f ∩  ( D ∗ ∩ ( p, q )) × ( E ∗ ∩ ( a, b ))  is uncoun table, a nd is an order-preserving bijection from D ∗ ∩ ( p, q ) o n to E ∗ ∩ ( a, b ). 4 EVER YWHERE D IFFERENTIABLE FUNCTIONS 8 No w fix s, t ∈ Q so that f ∗ ∩ ( ˜ H + s ) × ( ˜ K + t ) is uncoun table, so in particular it con tains a conv ergen t sequence. So, we hav e ( x n , y n ) ∈ f ∗ for n ≤ ω , with x n → x ω and y n → y ω as n ր ω , and x n ∈ ˜ H + s and y n ∈ ˜ K + t for all n ≤ ω . W e may assume that a ll the x n are distinct and that all the y n are distinct. Since f ∗ is order-preserving and the prop erty of H , K in Lemma 3.1 is preserv ed by translation, ∀ ε > 0 ∃ n ∈ ω [( y ω − y n ) / ( x ω − x n ) ∈ (0 , ε ) ∪ (1 / ε, ∞ )]. Pass ing to a subseque nce, w e may assume that either ∀ n ∈ ω [( y ω − y n ) / ( x ω − x n ) ∈ ( 2 n , ∞ ) ] or ∀ n ∈ ω [( y ω − y n ) / ( x ω − x n ) ∈ ( 0 , 2 − n )]. In the first case, f ′ ( x ω ) do esn’t exist and f is not Lipsc hitz on ( p, q ). In the second case, ( f − 1 ) ′ ( y ω ) do esn’t exist and f − 1 is not Lipsc hitz on ( a, b ). F or (3), rep eat the argumen t, now letting f ∗ b e the set of all ( d, f ( d )) suc h that p < d < q and f ′ ( d ) exists and f ′ ( d ) 6 = 0. K 4 Ev e rywhere Di ffe ren t iable F unct ions W e prov e here some lemmas to b e used in the pro of of Theorem 1.6, where w e shall construct the isomorphism f along with its deriv ativ e g . Definition 4.1 F or g : R → R , let k g k = sup {| g ( x ) | : x ∈ R } ∈ [0 , ∞ ] . Definition 4.2 D is the set of al l me asur a b l e g : R → R such that k g k < ∞ and g ( x ) = lim h → 0 1 h R x + h x g ( t ) dt for al l x . By this last condition, if f ( x ) = R x 0 g ( t ) dt , then f ′ ( x ) = g ( x ) for all x . Note that D is a Banach space with the sup norm k · k . Also, D contains all b ounded contin uous f unctions, and ev ery function in D is of Baire class 1; that is, a p oint wise limit of contin uous f unctions. Ho w ev er, man y Baire 1 functions, suc h as χ { 0 } , fail to b e in D . A function in D can b e ev erywhere discon tin uous; this has b een kno wn since the 1 890s; se e pp. 412–421 of Hobson [6] for references. Katznelson and Strom b erg [7] describ e a metho d for constructing suc h f unctions whic h we can em b ed into our forcing construction. Here w e summarize their metho d and make some minor additio ns to it. Definition 4.3 F or ψ : R → R an d a 6 = b : A V b a ψ = 1 b − a Z b a ψ ( x ) dx . Definition 4.4 Fix C > 1 . ψ : R → R has the C –av erage prop erty iff ψ is b ounde d an d c ontinuous, and ψ ( x ) ≥ 0 for al l x , and A V b a ψ ≤ C min( ψ ( a ) , ψ ( b )) whenever a 6 = b . L et AP C b e the set of a l l functions with the C –aver a ge pr op erty. 4 EVER YWHERE D IFFERENTIABLE FUNCTIONS 9 So, the av erage v alue of ψ on an in terv al is b ounded b y C times the v alue at either endp oint. Note that either ψ ( x ) > 0 for all x or ψ = 0 for a ll x . Also, AP C is closed under finite sums and uniform limits, and if ψ ∈ AP C then ( x 7→ αψ ( β x + γ )) ∈ AP C for all α, β , γ ∈ R with α ≥ 0. AP C clearly con tains all non-negativ e constant functions, but also, by [7], the function (1 + | x | ) − 1 / 2 has the 4–a v erag e prop erty ; see also Lemma 4.7 b elow . F unctions in AP C can b e used t o build functions in D b y: Lemma 4.5 Fix C > 1 . Assume that al l ψ j ∈ AP C . L et g ( x ) = P j ∈ ω ψ j ( x ) , and a ssume that g ( x ) < ∞ fo r al l x and k g k < ∞ . Then g ∈ D . Pro of. Fix x ∈ R and ε > 0. It is sufficien t to pro duce a δ > 0 suc h t ha t: ∀ h ∈ ( − δ, δ ) \{ 0 } : g ( x ) − 2 ε ≤ A V x + h x g ≤ g ( x ) + ( C + 1) ε . ( ∗ ) Let g m ( x ) = P j 0 suc h that | g m ( x ) − g m ( x + h ) | ≤ ε for all h ∈ ( − δ, δ ) \{ 0 } . Then, fix suc h an h , and w e v erify ( ∗ ). F or the first ≤ , use g ( x ) − 2 ε ≤ g m ( x ) − ε ≤ A V x + h x g m ≤ A V x + h x g . F or the second ≤ , note that for each n ≥ m , ( g n − g m ) ∈ AP C , and hence A V x + h x ( g n − g m ) ≤ C ( g n ( x ) − g m ( x )) ≤ C ε . Letting n ր ∞ , w e get A V x + h x ( g − g m ) ≤ C ε , so that A V x + h x g ≤ A V x + h x g m + C ε ≤ g m ( x ) + ( C + 1) ε ≤ g ( x ) + ( C + 1) ε . K T o v erify that the function (1 + | x | ) − 1 / 2 has the 4–av erage prop erty : Lemma 4.6 Supp ose that ψ : R → [0 , ∞ ) is a b o unde d me asur abl e function such that ψ ( x ) = ψ ( − x ) for al l x , ψ is de cr e asing for x > 0 , and A V b 0 ψ ≤ C ψ ( b ) fo r al l b > 0 . Then ψ ∈ AP 2 C . Pro of. W e must show t hat A V b a ψ ≤ 2 C min( ψ ( a ) , ψ ( b )) whenev er a 6 = b . By symmetry , there are only t w o cases: Case I: a < 0 < b , where 0 < ˆ a := − a ≤ b (so ψ ( a ) ≥ ψ ( b )): A V b a ψ = 1 b + ˆ a  Z b 0 ψ + Z ˆ a 0 ψ  ≤ 1 b · 2 Z b 0 ψ ≤ 2 C ψ ( b ) . Case I I: 0 ≤ a < b : Then, since ψ is decreasing, A V b a ψ ≤ A V b 0 ψ ≤ C ψ ( b ). K Lemma 4.7 If ψ ( x ) = (1 + | x | ) − 1 / 2 then ψ ∈ A P 4 . Pro of. F or b > 0, 1 ψ ( b ) A V b 0 ψ = √ 1 + b b h 2 √ 1 + b − 2 i = 2 b h b + 1 − √ 1 + b i < 2 , so a pply Lemma 4.6. K 5 ISOMORPHISMS 10 5 Isomorphis ms This en tire section is dev oted to the pro of of Theorem 1.6. W e plan to construct f along with g = f ′ , whic h will b e in D ; so f ( x ) = R x 0 g ( t ) dt . W e shall construct g as a limit of an ω –sequence, using the follow ing mo dification of Lemma 4.5: Lemma 5.1 Assume that we have g n , ψ n , θ n for n ∈ ω such that: 1. g 0 ∈ C ( R , [0 , ∞ ) ) and k g 0 k < ∞ . 2. θ n ∈ C ( R , R ) , and P n k θ n k < ∞ . 3. Each ψ n ∈ AP 4 . 4. g n +1 = g n − ψ n + θ n and g n +1 ( x ) ≥ 0 for al l x . Then h g n : n ∈ ω i c onver ges p ointwise to so me g : R → [0 , ∞ ) , and g ∈ D . Pro of. Since all ψ i ≥ 0 and all g n ≥ 0 , all sums h n := P i h t( e α i ) for al l α, i . c. ht( d α i ) 6 = h t( d α j ) for al l α , i, j with i 6 = j . Then ther e exist α 6 = β with p α 6⊥ p β , in the sense that for al l i < n , the slop e ( e β i − e α i ) / ( d β i − d α i ) > 0 and also | e β i − e α i | < ϕ ( | d β i − d α i | ) . Here, w e are a sserting that the t w o-elemen t partial function { ( d α i , e α i ) , ( d β i , e β i ) } is order- preservin g, and also has slop e b ounded b y a “small” function ϕ . Pro of. Induct on n . The case n = 0 is trivial, so assume the result for n and w e prov e it for n + 1, so now p α = (( d α 0 , e α 0 ) , . . . , ( d α n , e α n )) ∈ R 2 n +2 . Applying (b)(c), WLOG , each sequence is a rranged so that h t( p α ) = h t( d α n ), and hence h t( p α ) > ht( d α i ) for all i < n and h t( p α ) > ht( e α i ) for all i ≤ n . Also, by (a), WLOG, α < β → h t( p α ) < h t( p β ), whic h implies that ht( p α ) ≥ α . Let F = cl { p α : α < ω 1 } ⊆ R 2 n +2 . F or eac h α and each x ∈ R , o btain q α x ∈ R 2 n +2 b y replacing the d α n b y x in p α . Let F α = { x ∈ R : q α x ∈ F } . Fix ζ suc h that F ∈ M ζ . F or α ≥ ζ , F α is uncoun table b ecause d α n ∈ F α and F α ∈ M h t( p α ) while d α n / ∈ M h t( p α ) . So, c ho ose any u α , v α ∈ F α with u α < v α . Then, get an uncoun ta ble S ⊆ ω 1 \ ζ , alo ng with rationa l op en inte rv als U, V suc h that sup U < inf V and u α ∈ U a nd v α ∈ V . Let Ξ = inf { ϕ ( y − x ) : y ∈ V & x ∈ U } . Thinning S , we assume also that for α, β ∈ S , | e α n − e β n | < Ξ. Let ∗ p α = p α ↾ (2 n ) (delete the last pair). Applying induction, fix α, β ∈ S such that ∗ p α 6⊥ ∗ p β and e α n < e β n . No w, q α u α , q β v β ∈ F , so w e ma y c ho ose p δ , p ǫ sufficien tly close to q α u α , q β v β ∈ F , r esp ectiv ely , suc h that ∗ p δ 6⊥ ∗ p ǫ and also so that d δ n ∈ U and d ǫ n ∈ V , a nd also so that 0 < e ǫ n − e δ n < Ξ. Then ( e ǫ n − e δ n ) / ( d ǫ n − d δ n ) > 0 and also | e ǫ n − e δ n | < ϕ ( | d ǫ n − d δ n | ), so p δ 6⊥ p ǫ . K Our fo rcing conditions will contain, among other things, a finite σ ⊆ D × E whic h is a partial isomor phism; this σ will b e a sub-function of the f of Theorem 1.6. W e let g 0 ( x ) = x 2 / ( x 2 + 1), so that f 0 ( x ) = x − arctan( x ). The forcing condi- tions will determine success iv ely ψ 0 , θ 0 , ψ 1 , θ 1 , . . . , and hence also g 1 , f 1 , g 2 , f 2 , . . . . W e shall demand that all ψ n , θ n ∈ M 1 (and hence also all g n , f n ∈ M 1 ), so that there are only coun tably man y possibilities for them; this will f acilitate the pro of that t he p oset is ccc. Then, lim n f n = f ⊃ σ ; the f n will not actually extend σ ; rather, they will approximate σ in the sense of the following definition: Definition 5.4 ( τ , g , f , ι ) is corr ectable iff: ˘ P1. (0 , 0) ∈ τ and τ ∈ [ R × R ] <ω . ˘ P2. τ is a n or d e r- p r eserving bije c tion. 5 ISOMORPHISMS 12 ˘ P7. g ∈ C ( R , [0 , ∞ )) and g − 1 { 0 } = { 0 } . ˘ P8. f ( x ) = R x 0 g ( t ) dt . ˘ P13. ι > 0 and wh enever ( d 0 , e 0 ) , ( d 1 , e 1 ) ∈ τ and d 0 < d 1 : 0 < e 1 − e 0 d 1 − d 0 − f ( d 1 ) − f ( d 0 ) d 1 − d 0 < ι . The lab els on these items corresp ond to the lab els in Definition 5.6 (of P ). In P , the f , g will b e replaced by suitable f n , g n . Think of ι a s b eing “v ery small”. So, o ur hypotheses ( ˘ P2)( ˘ P3)( ˘ P7)( ˘ P13) imply that f and τ are strictly increasing, and b etw een d 0 , d 1 ∈ dom( τ ), the slop e of f is ve ry sligh tly less t ha n the slop e o f τ . W e remark that it is sufficien t to a ssume that ( ˘ P13) holds b et w een adjacen t elemen ts of dom ( τ ); that implies the full ( ˘ P13), since if d 0 < d 1 < d 2 w e hav e 0 <  τ ( d 1 ) − τ ( d 0 )  −  f ( d 1 ) − f ( d 0 )  < ι ( d 1 − d 0 ) & 0 <  τ ( d 2 ) − τ ( d 1 )  −  f ( d 2 ) − f ( d 1 )  < ι ( d 2 − d 1 ) = ⇒ 0 <  τ ( d 2 ) − τ ( d 0 )  −  f ( d 2 ) − f ( d 0 )  < ι ( d 2 − d 0 ) . Since f (0) = τ (0) = 0, w e can set d 0 = 0 or d 1 = 0 in ( ˘ P13) to obtain, for ( d, e ) ∈ τ : ˘ P12. d, e > 0 → 0 < ( e − f ( d )) < ιd ; d, e < 0 → 0 > ( e − f ( d )) > ιd . That is, if ( d, e ) ∈ τ , then f ( d ) is a sligh t under-estimate o f e when d > 0 and a sligh t ov er-estimate of e when d < 0. The next lemma sa ys that this “erro r” can b e corrected by adding a small p ositiv e function θ to g : Lemma 5.5 Assume that ( τ , g , f , ι ) is c orr e ctable and J ⊂ R is finite. Then for some θ : R → R : a. θ ( x ) ≥ 0 for al l x , and k θ k < ι , and θ ( x ) → 0 as x → ±∞ . b. θ is c o n tinuous, and θ ( d ) = 0 f or al l d ∈ J . c. If g ∗ = g + θ an d f ∗ ( x ) = R x 0 g ∗ ( t ) d t , then f ∗ ( d ) = e for e ach ( d, e ) ∈ τ . Pro of. Since τ (0) = f (0) = 0, item (c) will hold if w e hav e, for adja cent d 0 , d 1 ∈ dom( τ ) with d 0 < d 1 : Z d 1 d 0 θ ( t ) dt = ( τ ( d 1 ) − τ ( d 0 )) − ( f ( d 1 ) − f ( d 0 )) , and this quan tit y is assumed to lie in (0 , ι ( d 1 − d 0 )). It is no w easy to construct a C ∞ function θ whic h satisfies this, along with (a)(b). K 5 ISOMORPHISMS 13 Definition 5.6 P is the se t of al l tuples p = ( σ p , N p , g p n +1 , f p n +1 , ψ p n , θ p n ) n 0 → 0 < ( e − f N p ( d )) < 2 − N p − 2 d an d d, e < 0 → 0 > ( e − f N p ( d )) > 2 − N p − 2 d . P13. Whene v e r ( d 0 , e 0 ) , ( d 1 , e 1 ) ∈ σ and d 0 < d 1 : 0 < e 1 − e 0 d 1 − d 0 − f N p ( d 1 ) − f N p ( d 0 ) d 1 − d 0 < 2 − N p − 2 . Define q ≤ p iff Q1. σ q ⊇ σ p and N q ≥ N p . Q2. ( g p n +1 , f p n +1 , ψ p n , θ p n ) = ( g q n +1 , f q n +1 , ψ q n , θ q n ) for al l n < N p . Q3. Whene ver (0 , 0) 6 = ( d, e ) ∈ σ p and N p < n ≤ N q : g q n ( d ) ∈ (0 , 2 − n ) . Then 1 = ( { (0 , 0) } , 0) ; that is, whe n N p = 0 , the r est of the tuple is empty. W e shall now pro v e a sequence o f lemmas leading up t o Theorem 1.6, at the same time explaining some of the clauses in D efinition 5.6. The restriction on heigh ts in (P3)(P4) will b e imp ortan t in the pro of of ccc, and are ana lo gous to the restrictions in Lemma 5.3 . If G is a g eneric filter on P , then in V [ G ] w e can define b σ = S { σ p : p ∈ G } . Then b σ is an order-preserving function from a subset of D t o a subset of E , and the f of Theorem 1.6 will extend b σ (Lemma 5.10 b elo w). W e shall apply Lemma 5.1 in V [ G ] to obtain f , g , a nd (Q3) will let us pro v e that g ( d ) = 0 for a ll d ∈ dom( b σ ). Note that (Q3) is v acuous when N p = N q . 5 ISOMORPHISMS 14 By (P1)(P2)(P7)(P8)(P13), eac h ( σ, g N , f N , 2 − N − 2 ) is correctable. Then, as noted ab o v e, (P12) follow s, but we state it separately fo r emphasis, since it is used to pro v e that f extends b σ . Also, t he g n ( x ) < 2 − 2 − n asserted b y (P7) follow s b y induction from the other assumptions; sp ecifically , g 0 ( x ) < 1, g n +1 = g n − ψ n + θ n , θ n ( x ) ≤ 2 − n − 1 , and ψ n ( x ) ≥ 0. Definition 5.7 µ ( p ) = min {| d 0 − d 1 | : d 0 , d 1 ∈ dom( σ p ) & d 0 6 = d 1 } . Cal l a map ζ fr om P into the r ationals a P – function iff ζ ( p ) ∈ (0 , µ ( p ) / 2) for al l p . F or such a ζ , say p, q ∈ P ar e ζ – close iff N p = N q and | σ p | = | σ q | and ζ ( p ) = ζ ( q ) and al l elements of dom( σ p ) ∪ dom( σ q ) have differ ent he ights and ( g p n +1 , f p n +1 , ψ p n , θ p n ) n i } is dense in P for e ach i . So, in V [ G ], w e hav e g n , f n , ψ n , θ n for eac h n ∈ ω ; e.g., g n = g p n for some (an y) p ∈ G suc h that N p ≥ n . Th en Lemma 5.1 applies: (1) is obvious, (2) follo ws from (P11) , (3) follow s from (P10), and (4) follo ws fro m (P7)(P9), So, b y Lemma 5.1, h g n : n ∈ ω i con v erges p oin t wise to some g : R → [0 , ∞ ), and g ∈ D ; also, k g k ≤ 2 b y (P7). Then, since the g n are uniformly b ounded, h f n : n ∈ ω i con v erges p oin t wise to f , where f ( x ) = R x 0 g ( t ) dt . 5 ISOMORPHISMS 15 Regarding (Q3): By not r equiring g n ( d ) ≈ 0 for al l n ≤ N , w e make it easier to add new pairs ( d, e ) in to extensions of p (see the pro of of Lemma 5.13). Lik ewise , we only r equire (P12)(P13) f or n = N p , so that when pro ving Lemma 5.13, we do not need to consider (P12)(P13) for n < N p . But still, Lemma 5.10 F or ( d, e ) ∈ b σ : g ( d ) = 0 and f ( d ) = e . Pro of. Since h g n : n ∈ ω i and h f n : n ∈ ω i con v erge p oint wise, it is sufficien t to sho w that some subsequence of h g n ( d ) : n ∈ ω i conv erges to 0 and some subseque nce of h f n ( d ) : n ∈ ω i conv erges to e . Sa y ( d, e ) ∈ p ∈ G . Then b y Corollary 5.9, S := { N q : q ∈ G ∧ ( d, e ) ∈ q } is infinite. Then, applying (Q3)(P12), h g n ( d ) : n ∈ S i con v erges to 0 and h f n ( d ) : n ∈ S i con v erges to e . K Another consequence of Lemma 5.8: Lemma 5.11 P has the c c c. Pro of. Let A ⊆ P b e uncoun table; w e pro v e that A cannot b e a n antic hain. Let ζ ( p ) b e as in Lemma 5.8. W e may assume that ζ ( p ) is t he same rational ζ for all p ∈ A . F urthermore, by a delta system argumen t, we may assume that A = { p α : α < ω 1 } and σ p α = σ α ∪ τ , where τ is the ro ot of the delta syste m. W e may also assume (applying (P2)(P3)) t ha t the σ α satisfy the h yp otheses of Lemma 5.3 , and that all p α , p β satisfy ev erything in Definition 5.7 (of “ ζ –clos e”) except p ossibly for the requiremen t “ d 6 = d ′ implies 0 < ( e − e ′ ) / ( d − d ′ ) < ζ ”. But now Lemma 5.3 implies that there is some pair p α , p β with α 6 = β satisfying this requiremen t, so that p α 6⊥ q α b y Lemma 5.8. K By applying Lemma 5.5 to P w e g et: Lemma 5.12 Fix p ∈ P and a fini te F ⊂ R . L et N = N p and σ = σ p . T hen for some θ : R → R : a. θ ( x ) ≥ 0 for al l x , and k θ k < 2 − N − 2 , and θ ( x ) → 0 as x → ±∞ . b. θ is c o n tinuous, and θ ( d ) = 0 f or al l d ∈ F . c. If g ∗ = g N + θ and f ∗ ( x ) = R x 0 g ∗ ( t ) d t , then f ∗ ( d ) = e for e ach ( d, e ) ∈ σ . Lemma 5.13 ran( b σ ) = E . Pro of. It is sufficien t to pr ov e that for eac h e ∈ E , { q : e ∈ ran( σ q ) } is dens e. So fix p ∈ P with e / ∈ ran( σ p ), and we find a q ≤ p with e ∈ ra n( σ q ); q will b e exactly lik e p , except that σ q = σ p ∪ { ( d, e ) } , where d ∈ D ξ := { d ∈ D : h t( d ) = ξ } and 5 ISOMORPHISMS 16 h t( e ) < ξ < h t( e ) + ω a nd ξ is different fro m h t( d ′ ) for all d ′ ∈ dom( σ p ). Then q ≤ p is clear, but w e mus t mak e sure that q ∈ P . Let f ∗ b e as in Lemma 5.1 2 . Then f ∗ is a contin uous increasing function, and, using the lim x →±∞ g N ( x ) = 1 from (P7), f ∗ ( x ) → ∞ a s x → ∞ a nd f ∗ ( x ) → −∞ as x → −∞ . There is thus a unique ˆ d suc h that f ∗ ( ˆ d ) = e . F or all d sufficien tly close to ˆ d , setting σ q = σ p ∪ { ( d, e ) } will satisfy (P2)(P12)(P13) , so c ho ose suc h a d in D ξ , which is p ossible b ecause D ξ is dense. K Although dom ( b σ ) 6 = D (b y (P3) ( P4)), we do hav e: Lemma 5.14 In V [ G ] , dom( b σ ) is an ℵ 1 –dense subset of D . Pro of. Use the f acts that f is strictly increasing and contin uous, k f ′ k < ∞ (by P 7), f ⊃ b σ (b y L emma 5.10), and V , V [ G ] ha v e the same ℵ 1 (b y the ccc). K W e ar e no w done if w e prov e Lemma 5.8. First, a few remarks. As noted ab ov e, to prov e that p 6⊥ q whenev er p, q are ζ –close, w e need to mak e sure that the common extension satisfies (P2)(P12)(P13). But (P12) is a special case of ( P13), and it is easy to satisfy (P2); that is, if the function ζ is small enough then σ p ∪ σ q will b e order- preservin g. A more serious issue is that t he natural extension, ( σ p ∪ σ q , N , g n +1 , f n +1 , ψ n , θ n ) n 1 − p ζ . No w, assume that our f unction ζ ( p ) satisfies ∀ d ∈ dom( σ p ) ∀ x [ | x − d | < 2 ζ ( p ) → | g N ( x ) − g N ( d ) | < 2 − N / 256]. Then, using ¯ γ ℓ < γ ℓ : g N ( x ) − ψ N ( x ) ≤  γ ℓ + 2 − N / 256  −  γ ℓ + 2 − N / 16  1 − p ζ  when | x − ¯ d ℓ | < ζ , so that g N ( x ) − ψ N ( x ) ≤ 2 − N / 256 − 2 − N / 16 +  γ ℓ + 2 − N / 16  p ζ < 0 ; This la st < holds if w e assume that alwa ys p ζ ( p ) < 2 − N / 256. No w, we define θ † N ( x ) = ma x(0 , ψ N ( x ) − g N ( x )) + ε x 2 / ( x 2 + 1), where ε is a p ositiv e ra tional whic h is small enough to mak e the following a r gumen t work. Let g † N ( x ) := g N ( x ) − ψ N ( x ) + θ † N ( x ), whic h is p ositiv e ev erywhere except at 0. L et f † N ( x ) = R x 0 g † N ( t ) d t . W e plan to sho w that ( σ p ∪ σ q , g † N , f † N , 2 − N − 2 ) is correctable. (P11) requires k θ N k ≤ 2 − N − 1 . T o accomplish this, w e first v erify that k θ † N k ≤ 2 − N − 2 , and later w e shall v erify that k θ ∗ N k ≤ 2 − N − 2 . As long as ε ≤ 2 − N − 2 , 5 ISOMORPHISMS 18 θ † N ( x ) ≤ 2 − N − 2 whenev er ψ N ( x ) ≤ g N ( x ), which holds as x → ±∞ since g N ( x ) → 1 and ψ N ( x ) → 0. Also, θ † N ( x ) ≤ ψ N ( x ) + ε , so that if ε ≤ 2 − N − 3 , t hen θ † N ( x ) ≤ 2 − N − 2 whenev er ψ N ( x ) ≤ 2 − N − 3 , and if ζ is large enough, this will hold unless x is v ery close to one of the ¯ d ℓ . More precisely , if | x − ¯ d ℓ | ≥ c for all ℓ , then ψ N ( x ) ≤ 2 L  1 + r c  − 1 / 2 < 2 Lr − 1 / 2 c − 1 / 2 . Then, using 1 / √ ζ < r < 2 / √ ζ , if | x − ¯ d ℓ | ≥ ζ 1 / 4 for all ℓ then ψ N ( x ) < 2 Lζ 1 / 4 ζ − 1 / 8 = 2 Lζ 1 / 8 . Then θ † N ( x ) ≤ 2 − N − 2 for thes e x pro vided w e assume tha t our ζ function satisfies 2 | σ p | · ( ζ ( p )) 1 / 8 ≤ 2 − N p − 3 . No w, fix x and assume that | x − ¯ d m | ≤ ζ 1 / 4 for some m ; this m will b e unique if we assume tha t ( ζ ( p )) 1 / 4 < µ ( p ) / 4 for all p . W e need to sho w that θ † N ( x ) ≤ 2 − N − 2 . Assume that ψ N ( x ) > g N ( x ), since w e hav e already cov ered the case that g N ( x ) ≤ ψ N ( x ). So, θ † N ( x ) = ψ N ( x ) − g N ( x ) + εx 2 / ( x 2 + 1) ≤ ε +  γ m + 2 − N / 16  1 + r | x − ¯ d m |  − 1 / 2 + X ℓ 6 = m  γ ℓ + 2 − N / 16  1 + r | x − ¯ d ℓ |  − 1 / 2 − g N ( x ) ≤ ε +  γ m + 2 − N / 16  + 2 − N − 4 − g N ( x ) = ε + ( γ m − g N ( x )) + 2 − N − 2 / 2 ; for the last ≤ , use the previous argumen t, but no w assuming that our ζ function satisfies 2 | σ p | · ( ζ ( p )) 1 / 8 ≤ 2 − N p − 4 . Assuming that ε ≤ 2 − N − 2 / 4 and γ m − g N ( x ) ≤ 2 − N − 2 / 4 w e hav e θ † N ( x ) ≤ 2 − N − 2 . Since ¯ γ m ∈ { g N ( d p m ) , g N ( d q m ) , g N ( ¯ d m ) } and ¯ γ m < γ m < ¯ γ m + 2 − N / 256, and x is within 2 ζ 1 / 4 of eac h of d p m , d q m , ¯ d m , w e o bt a in γ m − g N ( x ) ≤ 2 − N − 2 / 4 if we assume that ∀ x ∀ d ∈ do m( σ p ) [ | x − d | ≤ 2 ( ζ ( p )) 1 / 4 → | g N ( x ) − g N ( d ) | ≤ 2 − N − 2 / 16. W e next sho w that ( σ p , g † N , f † N , 2 − N − 2 ) and ( σ q , g † N , f † N , 2 − N − 2 ) are correctable. T o do this, we b ound the change in f N ( d ) caused by replacing g N b y g N − ψ N + θ † N ; this c hange is R d 0 ( ψ N ( t ) − θ † N ( t )) d t . Let ∆ b e the diameter of dom( σ p ). Then, since γ ℓ + 2 − N / 16 < 2 and r > 1 / √ ζ Z d 0 ψ N ( t ) d t < 2 L Z ∆ 0 (1 + r t ) − 1 / 2 dt = 4 L r h √ 1 + r ∆ − 1 i ≤ 4 L r √ r ∆ ≤ 4 L √ ∆ 4 p ζ . This can b e made arbitra rily small b y requiring the ζ function to b e small enough. Lik ewise , R d 0 θ † N ( t ) d t can b e made arbitrarily small using 0 ≤ θ † N ( t ) ≤ ψ N ( t ) + ε . 5 ISOMORPHISMS 19 So, the correctabilit y of ( σ p , g † N , f † N , 2 − N − 2 ) and ( σ q , g † N , f † N , 2 − N − 2 ) follo ws fr o m the correctabilit y of ( σ p , g N , f N , 2 − N − 2 ) and ( σ q , g N , f N , 2 − N − 2 ) if ζ make s f † N close enough to f n . No w, to ve rify that ( σ p ∪ σ q , g † N , f † N , 2 − N − 2 ) is correctable, we must show that ( ˘ P13) holds b et w een a djacen t elemen t s of dom( σ p ∪ σ q ). There are t w o cases not already cov ered by the ab o v e: Case I: Bet w een d p m and d q ℓ where m 6 = ℓ : W e need 0 < e p m − e q ℓ d p m − d q ℓ − f † N ( d p m ) − f † N ( d q ℓ ) d p m − d q ℓ < 2 − N − 2 . This is handled by making ζ small enough, since the inequality holds if w e replace d q ℓ , e q ℓ b y d p ℓ , e p ℓ . Case I I: Bet w een d p ℓ and d q ℓ , when d p ℓ 6 = d q ℓ . WLOG, d p ℓ < d q ℓ , and w e need 0 < e q ℓ − e p ℓ d q ℓ − d p ℓ − f † N ( d q ℓ ) − f † N ( d p ℓ ) d q ℓ − d p ℓ < 2 − N − 2 . By the definition of “close”, w e hav e 0 < ( e q ℓ − e p ℓ ) / ( d q ℓ − d p ℓ ) < ζ , and o ur assumptions ab o v e abo ut ζ already imply that ζ < 2 − N − 2 . Th us, it is sufficien t to ha v e f † N ( d q ℓ ) − f † N ( d p ℓ ) < e q ℓ − e p ℓ . Now we hav e already c hec ked that g N ( x ) − ψ N ( x ) < 0 for x ∈ [ d p ℓ , d q ℓ ], so that g † N ( x ) = εx 2 / ( x 2 + 1) for these x . Then f † N ( d q ℓ ) − f † N ( d p ℓ ) = ε R d q ℓ d p ℓ x 2 / ( x 2 + 1) dx < ε ( d q ℓ − d p ℓ ), whic h will b e less tha n e q ℓ − e p ℓ if w e ha v e c hosen a small enough ε . Then, b y Lemma 5.5, there is a positive function θ # N suc h that k θ # N k < 2 − N − 2 and, setting g # N +1 = g † N + θ # N and in tegrating, giv es us f # N +1 ⊃ σ s ; so, instead of (P13) for s w e hav e, fo r ( d 0 , e 0 ) , ( d 1 , e 1 ) ∈ σ s and d 0 < d 1 : ( e 1 − e 0 ) − ( f # N +1 ( d 1 ) − f # N +1 ( d 0 )) = 0 . This is not exactly what we w an t, and this θ # N need not b e in M 1 , but b y mo difying our θ # N sligh tly , w e can get θ ∗ N ∈ M 1 so that setting g s N +1 = g † N + θ ∗ N and in tegrating giv es us f s N +1 satisfying 0 < ( e 1 − e 0 ) − ( f s N +1 ( d 1 ) − f s N +1 ( d 0 )) < 2 − N − 3 ( d 1 − d 0 ) , whic h is (P13) for the fo rcing condition s , so that s ∈ P . Of course, w e also need to verify that s ≤ p and s ≤ q . ( Q 1) and ( Q 2) are trivial, but (Q3) requires g s N +1 ( d ) ∈ (0 , 2 − N − 1 ) fo r d ∈ dom( σ p ) ∪ dom( σ q ) \{ 0 } . No w g s N +1 = g † N + θ ∗ N , and w e already kno w tha t g † N ( d ) = εd 2 / ( d 2 + 1) < ε , and w e already assumed tha t ε ≤ 2 − N − 2 . So, when w e apply Lemma 5.5, get θ # N ( d ) = 0 fo r these d . Then, when w e mo dify θ # N sligh tly to get θ ∗ N , make sure that θ # N ( d ) − θ ∗ N ( d ) ∈ (0 , 2 − N − 2 ). REFERENCES 20 References [1] U. Abraham, M. Rubin, and S. Shelah, On the consis tency of some partition theorems for con tin uous colorings, and the structure of ℵ 1 -dense real order t yp es, Ann. Pur e Appl. L o gic 29 (1985) 123-206. [2] U. Avraham and S. Shelah, Martin’s axiom do es not imply that eve ry tw o ℵ 1 -dense sets of reals are isomorphic, Isr ael J. Math. 38 (1981) 161 - 176. [3] J. Baumgartner, All ℵ 1 -dense sets of reals can b e isomorphic. F und. Math. 79 (19 73) 101-106 . [4] J. Baumgartner, Applications of the prop er forcing axiom, in Handb o ok of Set-The or etic T op olo gy , North-Holland, 1984, pp. 913-9 59. [5] J. Hart and K. Kunen, Arcs in the plane, to app ear, T op ol o gy Appl. [6] E. W. Hobson, The The ory of F unctions of a R e al V ariable and the T he ory of F ourier’s Series , V ol. 2, 2 nd Edition, The Univ ersit y Press, 1926. [7] Y. Katznelson and K. Strom b erg, Ev erywhere differen tiable, no where mono- tone, f unctions, Amer. Math. Monthly 81 (1 9 74) 349- 354. [8] W. Kubi ´ s a nd B. V ejnar, Co v ering a n uncoun table square b y countably man y con tin uous functions, a r Xiv:07 1 0.1402v3, 2009. [9] K. Kunen, Lo cally connected hereditarily Lindel¨ o f compacta, to app ear, T op olo gy Appl. [10] G. P eano, Sur une courb e, qui remplit toute une aire plane, Math. Ann. 36 (1890) 157 -160. [11] W. Sierpi ´ nski, Hyp oth` ese du Continu , Chelsea Publishing Compan y , 1956.