Knot Points of Typical Continuous Functions

Shingo SAITO

（斎藤新悟）

University College London

http://www.ucl.ac.uk/~ucahssa/

(Joint work with David Preiss)

START ↱

Outline

Part I Background
Part II Statement of Main Theorem
Part III Outline of Proof

Part I Background

⟲⊞ 2(↱)

Typical continuous functions

Work in I := [0, 1]. C(I) := {f: I ⟶ ℝ | f: continuous }

equipped with the supremum norm.

Definition A typical f ∈ C(I) satisfies a property P

⟺^{def.} {f ∈ C(I) | P holds } is residual (comeagre) in C(I).

Example A typical f ∈ C(I) is nowhere differentiable. What can we say about Dini derivatives of a

typical f ∈ C(I)?

⟲⊞ 3(↱)

Dini derivatives

Definition (Dini derivatives)

For f ∈ C(I) and x ∈ I,

D^{+} f(x) := lim sup_{y↘x} \frac{f(y) − f(x)}{y − x} ,
D_{+} f(x) := lim inf_{y↘x} \frac{f(y) − f(x)}{y − x} ,
D^{−}f(x) := lim sup_{y↗x} \frac{f(y) − f(x)}{y − x} ,
D_{−}f(x) := lim inf_{y↗x} \frac{f(y) − f(x)}{y − x} .

⟲⊞ 4(↱)

Dini derivatives of a typical f ∈ C(I)

Theorem (Jarník, 1933)

A typical f ∈ C(I) has the property that D^{+} f(x) = D^{−}f(x) = ∞ and

D_{+} f(x) = D_{−}f(x) = −∞

for a.e. x ∈ I.

f

Such a point x is called a knot point of f.

Knot points of a typical f ∈ C(I)

For f ∈ C(I), N(f) := {x ∈ I | x is NOT a knot point of f }. Jarník’s theorem asserts that

N(f) is null for a typical f ∈ C(I). In what sense of smallness is it true that

N(f) is small for a typical f ∈ C(I)?

Theorem of Preiss and Zajíček

Theorem (Preiss and Zajíček, unpublished)

For a σ-ideal I on I, T.F.A.E.:

(1) N(f) ∈ I for a typical f ∈ C(I);

(2) I ∩ K is residual in K.

Here

K := {K ⊂ I | K is closed }

equipped with the Vietoris topology. (Hausdorff metric)

Problem

Problem

Characterise those families A of subsets of I for which

N(f) ∈ A for a typical f ∈ C(I).

Part II

Statement of Main Theorem

An observation

Problem

Characterise A ⊂ P(I) for which N(f) ∈ A for a typical f ∈ C(I).

It is easy to see that N(f) is F_{σ} for all f ∈ C(I).

Problem

Characterise F ⊂ F_{σ} for which N(f) ∈ F for a typical f ∈ C(I).

Problem

Characterise F ⊂ F_{σ} for which N(f) ∈ F for a typical f ∈ C(I).

It is easy to see that {F ⊂ F_{σ} | N(f) ∈ F for a typical f ∈ C(I) }

is a σ-filter of F_{σ} sets.

Residuality of families of F_{σ} sets

Proposition (S)

For F ⊂ F_{σ}, T.F.A.E.:

(1) {(K_{n}) ∈ K^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F} is residual in K^{ℕ}.

(2) {(K_{n}) ∈ K_{↗}^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F} is residual in K_{↗}^{ℕ}.

Here

K_{↗}^{ℕ} := {(K_{n}) ∈ K^{ℕ} | K_{1} ⊂ K_{2} ⊂ ⋯ }.

We say that F is residual in F_{σ} if the above conditions hold.

⟲⊞ 12(↱)

Main theorem

Main Theorem If F is residual in F_{σ}, then N(f) ∈ F for a typical f ∈ C(I).

Conjecture

The converse is also true:

if N(f) ∈ F for a typical f ∈ C(I), then F is residual in F_{σ}.

Part III Outline of Proof

Residuality and Banach-Mazur game

X: a topological space, S ⊂ X. Players I and II alternately choose a nonempty

open set. They must choose a subset of the set chosen in the previous turn.

{I: \atop II:} U_{1}⊃ V_{1} ⊃U_{2}⊃ V_{1} ⊃⋯

Player II wins iff ⋂_{n=1}^{∞} U_{n} = ⋂_{n=1}^{∞} V_{n} ⊂ S.

Fact

Player II has a winning strategy ⟺ S is residual.

⟲⊞ 15(↱)

Outline of the proof

Let A := {(K_{n}) ∈ K_{↗}^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F}. We know that A is residual in K_{↗}^{ℕ}. We want to show that

{f ∈ C(I) | N(f) ∈ F}

is residual in C(I).

It suffices to show that X = {f ∈ C(I) | ∃(K_{n}) ∈ A N(f) = ⋃_{n=1}^{∞} K_{n}} is residual in C(I).

Outline of the proof

A is residual in K_{↗}^{ℕ}

⇓

Player II has a winning strategy in the Banach-Mazur game for A ⊂ K_{↗}^{ℕ}

⇓ ← similar to Proposition

Player II has a winning strategy in another game for A ⊂ K_{↗}^{ℕ}

⇓ ← difficult

Player II has a winning strategy in the Banach-Mazur game for X ⊂ C(I)

⇓

X is residual in C(I)

⟲⊞ 17(↱)

⟲  END  Bye  ⊞