Abbas Edalat

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Interval Derivative

• Abbas Edalat
• Imperial College London
• www.doc.ic.ac.uk/ae

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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.

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Non Continuity of the Derivative
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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?

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Dinis Derivates of a Function (1892)

• Upper derivative at x is defined as
• Lower derivative at x is defined as

• 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.

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Example
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Interval Derivative

• Let IR a,b a, b ? R ? R and consider
  (IR, ?) with R as bottom.

• The interval derivative of f c,d ? R is
  defined as

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Examples
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Example

• We have already seen that

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Envelop of Functions

• The envelop of f is defined as

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Examples
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Envelop of Interval-valued Functions

• The envelop of f is now defined as

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

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

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

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Continuity of the Interval Derivative

• Theorem. The interval derivative of f c,d ?
  R is

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Computational Content of the Interval Derivative

• Definition. (AE/AL in LICS02) 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 .

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• Theorem. For f c,d ? R we have

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Fundamental Theorems of Calculus
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Interval Derivative, Ordinary Derivative and the
Lebesgue Integral
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Primitive of a Scott Continuous Map

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

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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.

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Construction of a Total Splitting

• Construct a fat Cantor set in 0,1 with

0
1
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Primitive of a Scott Continuous Function

• To construct with
  for a given

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How many primitives are there?

• Theorem. Given
  , for any
• with , there exists a total
  splitting A such that

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Fundamental Theorem of Calculus Revisited
derivative

• Continuously differentiable
• functions

Continuous functions
Riemann integral

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

• Primitives here differ by non-equivalent total
  splittings

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Higher Order Interval Derivatives.

• Extend the interval derivative to interval-valued
  functions

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Conclusion

• 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 ENDhttp//www.doc.ic.ac.uk/ae
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Locally Lipschitz functions

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

• 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