Abbas Edalat Imperial College London Interval Derivative

The Classical Derivative Let f: [a,b]  R be a real-valued function. The derivative of f at x is defined as when the limit exists (Cauchy 1821). If the derivative exists at x then f is continuous at x. 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 Derivative The derivative of f may exist in a neighbourhood O of x but the function may be discontinuous at x, e.g. with f(0)=0 we have: does not exist.

4 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) 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

6Example

7 Interval 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. if both limits are finite otherwise

8Examples

9Example with f(0)=0 We have already seen that We have Thus

10 Envelop of Functions Let and be any extended real-valued function. Then The envelop of f is defined as

11Examples

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

13 Also called upper continuity in set-valued function theory. Proposition. For any the envelop is continuous with respect to the Scott topology on IR. ⊑ ), 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. Thus env(f) is the computational content of f. For any Scott continuous with we have g ⊑ env(f).

14 Continuity of the Interval Derivative Theorem. The interval derivative of f: [c,d]  R is Corollary. is Scott continuous. Theorem. (i) If f is differentiable at x then (ii) If f is continuously differentiable in a neighboorhood of x then

15 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 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. are respectively lower and upper Lipschitz constants for f in the interval a. (x, f(x)) b b a Graph(f).

16 Theorem. For f: [c,d]  R we have: Recall the single-step function. If, b ax

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

18 Interval Derivative, Ordinary Derivative and the Lebesgue Integral Theorem. If is absolutely continuous then f: [c,d]  R ⊑ Theorem. If is absolutely continuous and is Lebesgue integrable with then f: [c,d]  R g: [c,d]  R

19 Primitive of a Scott Continuous Map Given Scott continuous is there with 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 ?

20 Total Splitting of an Interval A total splitting of [0,1] is given by a measurable subset such that for any interval we have: where is the Lebesgue measure. It follows that A and [0,1]\A are both dense with empty interior. A non-example:

21 Construction of a Total Splitting Construct a fat Cantor set in [0,1] with 01 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

22 Primitive of a Scott Continuous Function To construct with for a given Take any total splitting A of [0,1] and put Theorem. Let, then For and any total splitting A, cannot be made continuous by removing any null set.

23 How many primitives are there? Theorem. Given, for any with, there exists a total splitting A such that Let be the set of jumps of g. Then B is measurable and: Theorem. Two total splittings give rise to the same primitive iff their intersections with B are the same up to a null set:

24 Fundamental Theorem of Calculus Revisited Continuously differentiable functions Continuous functions derivative 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

25 Higher Order Interval Derivatives. Extend the interval derivative to interval-valued functions: The interval derivative of f: [c,d]  IR is defined as If f maps a neighbourhood O of y into real numbers with the induced function: Inductively define:

26Conclusion 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.

27 THE END

28 Locally Lipschitz functions A map f: (c,d)  R is locally Lipschitz if it is Lipschitz in a neighbourhood of each. 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