Interval Derivative of Functions

Interval Derivative of Functions Abbas Edalat Imperial College London www.doc.ic.ac.uk/~ae

The Classical Derivative when the limit exists (Cauchy 1821). • If the derivative exists at x then f is continuous at x. • However, a continuous function may not be differentiable at a point x and there are indeed continuous functions which are nowhere differentiable, the first constructed by Weierstrass: Let f: [a,b]  Rbe a real-valued function. The derivative of f at x is defined as with 0 <b< 1 and a an odd positive integer.

Non Continuity of the Derivative with f(0)=0 • The derivative of f may exist in a neighbourhood O of x but the function may be discontinuous at x, e.g. we have:

A Continuous Derivative for Functions? • A computable function needs to be continuous with respect to the topology used for approximation. • Can we define a notion of a derivative for real valued functions which is continuous with respect to a reasonable topology for these functions?

Dini’s Four Derivates of a Function (1890) • Upper right derivate at x • Upper left derivate at x • Lower right derivate at x • Lower left derivate at x Clearly, f is differentiable at x iff its four derivates are equal, the common value will then be the derivative of f at x.

Example

Interval Derivative • Put Let IR={ [a,b] | a, b  R}  {R}and consider (IR, ) with R as bottom. The interval derivative of f: [c,d]  Ris defined as if both limits are finite otherwise

Example with f(0)=0 • We have already seen that We have Thus

Interval-Limit of Functions Let and be any extended real-valued function. Then The interval limit of f is defined as

Examples

Interval-limit of Interval-valued Functions • Let and be extended real-valued functions with . • Consider the interval-valued function The interval-limit of f is now defined as

Continuity of the Interval Derivative Theorem. Given any the interval limit is Scott continuous. Corollary. The interval derivative of f: [c,d]  R is Scott continuous.

Computational Content of the Interval Derivative • Definition. (AE/AL in LICS’02) We say f: [c,d]  R has interval Lipschitz constant in an open intervalif The set of all functions with interval Lipschitz constantbatais called the tie of a with b and is denoted by . Recall the definition of single-step function Theorem. For f: [c,d]  R we have:

Fundamental Theorems of Calculus Continuous function versus continuously differentiable function for continuous f for continuously differentiable F Lebesgue integrable function versus absolutely continuous function for any Lebesgue integrable f iff F is absolutely continuous

Locally Lipschitz functions The interval derivative induces a duality between locally Lipschitz maps versus bounded integral functions and their interval limits. • A map f: (c,d)  R is locally Lipschitz if it is Lipschitz in a neighbourhood of each . A locally Lipschitz map f is differentiable a.e. and The interval derivative of a locally Lipschitz map is never bottom:

Primitive of a Scott Continuous Map Given Scott continuous is there with In other words, does every Scott continuous function has a primitive with respect to the interval derivative? For example, is there a function f with ?

Total Splittings of Intervals • A total splitting of [0,1] is given by disjoint measurable subsets A, B with such that for any interval we have: where is the Lebesgue measure. It follows that A and B are both dense with empty interior. Non-example:

Construction of a Total Splitting 0 1 • Construct a fat Cantor set in [0,1] with • In the open intervals in the complement of construct countably many Cantor sets with • In the open intervals in the complement of construct Cantor sets with . Continue to construct with Put

Primitive of a Scott Continuous Function • To construct with for a given Take any total splitting (A,B) of [0,1] and put Theorem.

Non-smooth Mathematics Smooth Mathematics • Geometry • Differential Topology • Manifolds • Dynamical Systems • Mathematical Physics . . All based on differential calculus • Set Theory • Logic • Algebra • Point-set Topology • Graph Theory • Model Theory . .

A Domain-Theoretic Model for Differential Calculus Indefinite integral of a Scott continuous function Derivative of a Scott continuous function Fundamental Theorem of Calculus for interval-valued functions Domain of C1 functions Domain of Ck functions

Continuous Scott Domains • A directed complete partial order (dcpo) is a poset (A, ⊑) , in which every directed set {ai |iI }  A has a sup or lub ⊔iI ai • The way-below relation in a dcpo is defined by: a ≪ b iff for all directed subsets {ai |iI }, the relation b ⊑ ⊔iI ai implies that there exists i Isuch that a ⊑ ai • If a ≪ b then a gives a finitary approximation to b • B A is a basis if for each a  A , {b  B | b ≪ a } is directed with lub a • A dcpo is(-)continuous if it has a (countable) basis • The Scott topology on a continuous dcpo A with basis B has basic open sets {a  A | b ≪ a } for each b  B • A dcpo is bounded complete if every bounded subset has a lub • A continuous Scott Domain is an -continuous bounded complete dcpo

The Domain of nonempty compact Intervals of R  {x} x x  {x} : R  IRTopological embedding R I R • Let IR={ [a,b] | a, b  R}  {R} • (IR, ) is a bounded complete dcpo with R as bottom: ⊔iI ai = iI ai • a ≪ b  ao  b • (IR, ⊑) is -continuous: countable basis {[p,q] | p < q & p, q  Q} • (IR, ⊑) is, thus, a continuous Scott domain. • Scott topology has basis:↟a = {b | ao  b}

Continuous Functions • f : [0,1]  R, f  C0[0,1], has continuous extension If : [0,1]  IR x  {f (x)} Scott continuous maps [0,1]  IR with:f ⊑ g x  R . f(x) ⊑ g(x)is another continuous Scott domain.  : C0[0,1] ↪ ( [0,1]  IR), with f  Ifis a topological embedding into a proper subset of maximal elements of [0,1]  IR .

Step Functions • Single-step function: a↘b : [0,1]  IR, with a  I[0,1], b  IR: b x ao x otherwise • Lubs of finite and bounded collections of single- step functions ⊔1in(ai ↘bi) are called step function. • Step functions with ai, birational intervals, give a basis for [0,1]  IR

Step Functions-An Example R b3 a3 b1 b2 a1 a2 0 1

Refining the Step Functions R b3 a3 b1 a1 b2 a2 0 1

Operations in Interval Arithmetic • For a = [a, a]  IR, b = [b, b]  IR,and *  { +, –,  } we have: a * b = { x*y | x  a, y  b } For example: • a + b = [ a + b, a + b]

The Basic Construction • Classically, with • What is the indefinite integral of a single step function a↘b ? • Intuitively, we expect f to satisfy: We expect  a↘b ([0,1]  IR) For what f  C1[0,1], should we have If   a↘b ?

Interval Derivative Assume f  C1[0,1], a  I[0,1], b  IR. Supposex  ao . b f (x)  b. We think of [b, b] as an interval derivative for f at a. • Note thatx  ao . b f (x)  b iff x1, x2 ao & x1 > x2 , b(x1 – x2)  f(x1) – f(x2)  b(x1 – x2), i.e. b(x1 – x2) ⊑ {f(x1) – f(x2)} = {f(x1)} – {f(x2)}

Definition of Interval Derivative • Proposition. For f: [0,1]  IR, we have f (a,b) iff f(x) Maximal (IR) forx  ao(hencef continuous)andGraph(f) is within lines of slopeb & b at each point (x, f(x)), x  ao. . b (x, f(x)) Graph(f) b a • f  ([0,1]  IR) has an interval derivativeb  IR at a  I[0,1] ifx1, x2 ao, b(x1 – x2) ⊑ f(x1) – f(x2). • The tie ofawithb, is (a,b) := { f | x1,x2  ao. b(x1 – x2) ⊑ f(x1) – f(x2)}

For Classical Functions Let f  C1[0,1]; the following are equivalent: • If (a,b) • x  ao . b f (x)  b • x1,x2 [0,1], x1,x2  ao. b(x1 – x2) ⊑ If (x1) – If (x2) • a↘b ⊑ If Thus, (a,b) is our candidate for a↘b .

Properties of Ties • (a1,b1) (a2,b2) iff a2⊑ a1& b1⊑ b2 • ni=1 (ai,bi) iff{ai↘bi | 1 i  n} bounded. • iI(ai,bi) iff {ai↘bi | iI } bounded iff J finite I iJ(ai,bi)  • In fact, (a,b) behaves like a↘b; we call (a,b) asingle-step tie.

The Indefinite Integral •  : ([0,1]  IR)  (P([0,1]  IR),  ) ( Pthe power set constructor) •  a↘b :=(a,b) •  ⊔i Iai ↘bi := iI(ai,bi) •  is well-defined and Scott continuous. • But unlike the classical case, the indefinite integral  is not 1-1.

Example • ([0,1/2] ↘ {0})⊔ ([1/2,1] ↘ {0}) ⊔ ([0,1] ↘ [0,1]) = ([0,1/2] , {0}) ([1/2,1] ↘ {0})  ([0,1] ↘ [0,1]) = ([0,1] , {0}) =  [0,1] ↘ {0}

The Derivative • Definition. Given f : [0,1]  IR the derivative of f is: : [0,1]  IR = ⊔{a↘b | f (a,b) } • Theorem. (Compare with the classical case.) • is well–defined & Scott continuous. • If f  C1[0,1], then • f (a,b) iff a↘b ⊑

Examples

The Derivative Operator • : ([0,1]  IR)  ([0,1]  IR)is monotone but not continuous. Note that the classical operator is not continuous either. • (a↘b)= x .  • is not linear! Forf : x  {|x|} : [0,1]  IR g : x  {–|x|} : [0,1]  IR (f+g) (0) (0) +(0)

Domain of Ties, or Indefinite Integrals • LetT[0,1] = Image ( ), i.e.  T[0,1] iff x is the nonempty intersection of a family of single-ties: =iI (ai,bi)    • Recall  : ([0,1]  IR)  (P([0,1]  IR),  ) • Domain of ties: ( T[0,1],  ) • Theorem.( T[0,1],  ) is a continuous Scott domain.

The Fundamental Theorem of Calculus • Theorem. : (T[0,1], )  ([0,1]  IR) is upper adjointto  : ([0,1]  IR) (T[0,1], ) In fact,Id = ° and Id ⊑ ° • Define : (T[0,1] , )  ([0,1]  IR) ∆ ⊓ { | f  ∆ }

Fundamental Theorem of Calculus ≡ For f, g  C1[0,1], let f ~ g if f = g + r, for some r  R. We have:

F.T. of Calculus: Isomorphic version ≡ ≡ • For f , g  [0,1]  IR, let f ≈ g if f = g a.e. • We then have:

A Domain for C1Functions • If h C1[0,1] , then ( Ih , Ih )  ([0,1]  IR)  ([0,1]  IR) • We can approximate ( Ih, Ih ) in ([0,1]  IR)2 i.e. ( f, g) ⊑ ( Ih ,Ih ) with f ⊑ Ih and g ⊑ Ih • What pairs ( f, g)  ([0,1]  IR)2approximate a differentiable function?

Function and Derivative Consistency • Proposition (f,g)  Cons iff there is a continuous h: dom(g) R with f ⊑ Ih and g ⊑ . • Define the consistency relation:Cons  ([0,1]  IR)  ([0,1]  IR) with(f,g)  Cons if (f)  ( g)  In fact, if(f,g)  Cons, there are least and greatest functionshwith the above properties in each connected component ofdom(g)which intersectsdom(f) .

Consistency for basis elements G(f,g)= greatest function fg(t) Approximating function: f = ⊔iai↘bi L(f,g) = least function t Approximating derivative: g = ⊔j cj↘dj (⊔iai↘bi, ⊔j cj↘dj)  Cons is a finitary property: • We will define L(f,g), G(f,g) in general and show that: • (f,g)  Cons iff L(f,g) G(f,g). • Cons isdecidable on the basis. Up(f,g) := (fg , g) where fg : t  [ L(f,g)(t) , G(f,g)(t) ]

Function and Derivative Information f 1 1 g 2 1

Updating f 1 1 g 2 1

Consistency Test and Updating for (f,g) • For x  dom(g), let g({x}) = [g (x), g+(x)] where g , g+:dom(g) R are lower and upper semi-continuous. Similarly we define f , f +:dom(f) R. Writef = [f –, f +]. Let O be a connected component of dom(g) with O  dom(f) . For x , y  O define: Define: L(f,g)(x) := supyOdom(f)(f –(y) + d–+(x,y))andG(f,g)(x) := infyOdom(f)(f +(y) + d+–(x,y)) Theorem. (f, g)  Con iffx  O. L (f, g) (x)  G (f, g) (x).

Updating Linear step Functions • A linear single-step function: a↘[b–, b +] : [0,1]  IR, withb–, b +: ao  Rlinear [b–(x) , b +(x)] x  ao x otherwise We write this simply as a↘b with b=[b–, b +] . • Proposition. For x  O, we have:L(f,g)(x) = max {f –(x) , limsup f –(y) + d–+(x , y) | ym  O  dom(f) } • For (f, g) = (⊔1inai↘bi , ⊔1jmcj↘dj) with f linear g standard, the rational end–points of aiand cj induce a partition y0 < y1 < y2 < … < ykof the connected component O of dom(g). • Hence L(f,g) is the max of k+2 linear maps. • Similarly for G(f,g)(x).

Updating Algorithm f 1 1 g 2 1

Interval Derivative of Functions