Abbas Edalat

Imperial College London

www.doc.ic.ac.uk/~ae

Interval DerivativeInterval Derivative

The Classical DerivativeThe Classical Derivative

• Let f: [a,b] R be a real-valued function.

The derivative of f at x is defined as

yx

f(y)f(x)lim(x)' f xy

when the limit exists (Cauchy 1821).

• If the derivative exists at x then f is continuous at x.

) x π(a cosb (x) f n

0n

n

with 0 <b< 1 and a an odd positive integer.

• 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:

3

Non Continuity of the DerivativeNon Continuity of the Derivative

• The derivative of f may exist in a neighbourhood O of x but the function

R O :' f

may be discontinuous at x, e.g.

12 xsin x x: f with f(0)=0

we have:

does not exist.

(x) ' f lim

0)(x xcossin x x 2(x)' f

0(0)' f

0x

11

4

A Continuous Derivative for Functions?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?

5

Dini’s Derivates of a Function (1892)Dini’s Derivates of a Function (1892)

• f is differentiable at x iff its upper and lower derivatives are equal, the common value will then be the derivative of f at x.

• Upper derivative at x is defined as

• Lower derivative at x is defined as yx

f(y)f(x)lim: (f)(x)D

yx

f(y)f(x)lim : (f)(x)D

xy

xyu

l

6

ExampleExample

1 (f)(0)D

1 (f)(0)Du

l

0 x 0

0 xsin xx x:f

1

0)(y ysin y

f(y)

y0

f(y)f(0) 1

7

Interval DerivativeInterval Derivative

• The interval derivative of f: [c,d] R is defined as

• Let IR={ [a,b] | a, b R} {R} and consider (IR, ) with R as bottom.

IR d][c, :dx

df

] (f)(y)Dlim , (f)(y)Dlim [

dx

df uxyxy

l if both limits are finiteotherwise

8

ExamplesExamples

0 x

0 x(x)}f{ x

IRR:dx

df

0f(0)with

RR:)sin(xx x:f 1

x

x

xx

0 {1}

0 ]1,1[

0 x}1{

x

IRR:dx

df

RR|:|:f

9

ExampleExample12 xsin x x: f with f(0)=0

0)(x xcossin x x 2(x)' f

0(0)' f11

• We have already seen that

• We have 0)(x (x)' f (f)(x)D (f)(x)Du l

1,1][ ] (x)' flim , (x)' flim [

] (f)(x)Dlim , (f)(x)Dlim [ )0(dx

df

0x0x

u0x0x

l

• Thus

10

Envelop of FunctionsEnvelop of Functions

• Let and be any extended real-valued function.RA:f RA

• ThenRA:f lim RA:f lim

• The envelop of f is defined as

A \ A x f(x)] lim , f(x) lim [

A x f(x))]f(x), lim max( , f(x))f(x), lim [min(

at x finitenot is f limor f lim f, if

IRA:env(f)

x

11

ExamplesExamples

0 xif 1

0 xif 1 x

R(0,1]1,0)[:f

0 xif 1,1][

0 xif {1}

0 xif 1}{

x

IR1,1][:env(f)

1sin x x

R)(0,,0)(:g

0 xif 1,1][

0 xif }{sin x x

IRR :env(g)1

12

Envelop of Interval-valued FunctionsEnvelop of Interval-valued Functions

A A x (x)]f lim , (x)f lim [

A x (x))]f(x),f lim max( , (x))f(x),f lim [min(

at x finitenot are f limor f lim ,f,f if

IRA:env(f)

x

otherwise

finite are (x)f , (x)f if (x)]f(x),[f f(x)

• The envelop of f is now defined as

• Let and be extended real-valued functions

with .

RA RA:f,f

ff

IRA:]f,[ff • Consider the interval-valued function

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• Also called upper continuity in set-valued function theory.

• Proposition. For any the envelop is continuous with respect to the Scott topology on IR.

IRA:env(f)

• ⊑), the collection of Scott continuous maps, ordered pointwise and equipped with the Scott topology, is a continuous Scott domain that can be given an effective structure.

IR,A(

IRA:]f,[ff

• Thus env(f) is the computational content of f.

• For any Scott continuous with

we have g env(f).⊑

IRA:]g,[gg

gf,fg

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Continuity of the Interval DerivativeContinuity of the Interval Derivative

• Theorem. The interval derivative of f: [c,d] R is

]) (f)D , (f)D [ (envdx

df ul

• Corollary. is Scott continuous. IR d][c, :dx

df

• Theorem. (i) If f is differentiable at x then x)(dx

df (x)' f

(x)}' {f(x)dx

df

• (ii) If f is continuously differentiable in a neighboorhood of x then

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Computational Content of the Interval DerivativeComputational Content of the Interval Derivative

• Definition. (AE/AL in LICS’02) We say f: [c,d] R has interval Lipschitz constant in an open interval if

The set of all functions with interval Lipschitz constant b at a is called the tie of a with b and is denoted by .

]b , b[b d][c,a

y)(xb f(y)f(x) y)(xb x.y,ayx,

b)δ(a,

• are respectively lower and upper Lipschitz constants for

f in the interval a.

b , b

(x, f(x))

b

b

a

Graph(f).

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• Theorem. For f: [c,d] R we have:

b)}δ(a,f& ay|{b y)(dx

df

:Thus

. b)}δ(a,f|b sup{adx

df step

• Recall the single-step function. If ,

otherwise

a xif b x with IRd][c,:ba step

b

ax

]b , b[b d][c,]a,a[a

17

Fundamental Theorems of CalculusFundamental Theorems of Calculus

• Continuous function versus continuously differentiable function (Riemann)

x

c

x

c

F(c)F(x) (t)dt F'

f(x) f(t)dt)'( for continuous f

for continuously differentiable F

• Lebesgue integrable function versus absolutely continuous function (Lebesgue)

x

c

x

c

F(c)F(x) (t)dt F'

a.e. f(x) f(t)dt)'( for any Lebesgue integrable f

iff F is absolutely continuous

ε|)F(x)F(y|δ)x(y

.yx...yxy x 0.δ . 0ε i.e.n

1i

n

1i iiii

nn2211

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Interval Derivative, Ordinary Derivative and the Interval Derivative, Ordinary Derivative and the Lebesgue IntegralLebesgue Integral

• Theorem. If is absolutely continuous thenf: [c,d] R

)' f (envdx

df

dx

df⊑

• Theorem. If is absolutely continuous and is Lebesgue integrable with

then

f: [c,d] Rg: [c,d] R

x

c g(t)dt

dx

d

)dx

df(g)

dx

df(

19

Primitive of a Scott Continuous MapPrimitive of a Scott Continuous Map

• Given Scott continuous is there

IR[0,1]:]g,[gg

R[0,1]:f

with

gdx

df

• In other words, does every Scott continuous function have a primitive with respect to the interval derivative?

• For example, is there a function f with ?]1,0[ ]1,0[dx

df step

20

Total Splitting of an IntervalTotal Splitting of an Interval

• A total splitting of [0,1] is given by a measurable subset such that for any interval

we have:q)(p [0,1]q][p,

0 A))\([0,1]q]λ([p, & 0 A)q]λ([p,

where is the Lebesgue measure.λ

• It follows that A and [0,1]\A are both dense with empty interior.

• A non-example: [0,1] QA

[0,1]A

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Construction of a Total SplittingConstruction of a Total Splitting

• Construct a fat Cantor set in [0,1] with

0 10C 1/4)λ(C0

0C• In the open intervals in the complement of construct

countably many Cantor sets with 8/1)Cλ(0i 0i 0)(i C0i

• In the open intervals in the complement of construct

Cantor sets with .

0i 0iC

0)(i C1i 16/1)Cλ(0i 1i

• Continue to construct with 0i jiC

j2

0i ji 1/2)Cλ(

• Put 1/2A)\λ([0,1]λ(A) withCA

0ji, ji

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Primitive of a Scott Continuous Function Primitive of a Scott Continuous Function

• To construct with for a given

IR[0,1]:]g,[gg

gdx

df R[0,1]:f

• Take any total splitting A of [0,1] and put

A xif (x)g

A xif (x)g (x)h R[0,1]:h AA

gdx

df • Theorem. Let , then

x

0 A (t)dth f(x)

• For and any total splitting A, cannot be made continuous by removing any null set.

]1,0[ ]1,0[]g,[g stepAh

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How many primitives are there?How many primitives are there?

• Theorem. Given , for any

with , there exists a total splitting A such that

IR[0,1]:]g,[gg

gdx

df

R[0,1]:f

x

0 A (t)dth f(x)

• Let be the set of jumps of g.

Then B is measurable and:

)}(g)(g |{xB xx

• Theorem. Two total splittings give rise to the same primitive iff their intersections with B are the same up to a null set:

0B)AB,λ(Δ(A (t)dth (t)dth 21

x

0 A

x

0 A 21

21 A,A

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Fundamental Theorem of Calculus RevisitedFundamental Theorem of Calculus Revisited

Continuously differentiable functions

Continuous functionsderivative

Riemann integral

Absolutely continuous functions

Lebesgue integrable functions

derivative

Lebesgue integral

• In both cases above, primitives differ by an additive constant

Scott continuous functions

Absolutely continuous functions

interval derivative

Lebesgue integral wrt total splittings

• Primitives here differ by non-equivalent total splittings

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Higher Order Interval Derivatives.Higher Order Interval Derivatives.

• Extend the interval derivative to interval-valued functions:

• The interval derivative of f: [c,d] IR is defined as

IR d][c, :dx

df

)(dx

fd

yy

If f maps a neighbourhood O of y into real numbers with the induced function:

{f(x)}(x) f R,O : f

• Inductively define: )dx

fd(

dx

d

dx

fd IR, d][c, :

dx

fdn

n

1n

1n

1n

1n

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ConclusionConclusion

• The interval derivative provides a new, computational approach, to differential calculus.

• It is a great challenge to use domain theory to synthesize differential calculus and computer, in order to extract the computational content of smooth mathematics.

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THE ENDTHE END

http://www.doc.ic.ac.uk/~aehttp://www.doc.ic.ac.uk/~ae

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Locally Lipschitz functions Locally Lipschitz functions

• A map f: (c,d) R is locally Lipschitz if it is Lipschitz in a neighbourhood of each .d)(c,y

(y)dx

df d).(c,y withIRd)(c,:

dx

df

• The interval derivative induces a duality between locally Lipschitz maps versus bounded integral functions and their envelops.

• The interval derivative of a locally Lipschitz map is never bottom:

• A locally Lipschitz map f is differentiable a.e. and

x

c f(c)f(x) (t)dt ' f