Typical F_{σ} Sets and Typical Continuous Functions

(\begin{matrix}Knot Points of \\ Typical Continuous Functions \end{matrix} )

Shingo SAITO

（斎藤新悟）

University College London

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

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Outline

Part I Background
Part II Statement of Main Theorem
Part III Sketch 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 (f)

⟺^{def.} {f ∈ C(I) | P (f) holds } is residual 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Theorem (Preiss and Zajíček, unpublished)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 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).

Main Theorem

Main Theorem (S)

For F ⊂ F_{σ}, T.F.A.E.:

(1) N(f) ∈ F for a typical f ∈ C(I);

(2) F is residual in F_{σ} (F ∈ F for a typical F ∈ F_{σ}).

What does residual mean in this context?

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(↱)

Part III Sketch of Proof

Statement of main theorem

Main Theorem

For F ⊂ F_{σ}, T.F.A.E.:

(1) N(f) ∈ F for a typical f ∈ C(I);

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

Proof of Main Thm

Lemma We may find a ‘good’ 𝕏 ⊂ K_{↗}^{ℕ} × C(I) s.t.

• ((K_{n}), f) ∈ 𝕏 implies ⋃_{n=1}^{∞} K_{n} = N(f);

• if A ⊂ K_{↗}^{ℕ} is residual, then

for a typical f ∈ C(I), ∃(K_{n}) ∈ A ((K_{n}), f) ∈ 𝕏.

The proof of this lemma

• is very complicated and

• uses the Banach-Mazur game.

(2) ⇒ (1)

Suppose

A := {(K_{n}) ∈ K_{↗}^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F}

is residual.

By Lem, for a typical f ∈ C(I), ∃(K_{n}) ∈ A ((K_{n}), f) ∈ 𝕏,

∴ N(f) = ⋃_{n=1}^{∞} K_{n} ∈ F.

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

⇒ (1) N(f) ∈ F for a typical f.

Lem 𝕏 ⊂ K_{↗}^{ℕ} × C(I) satisfies

• ((K_{n}), f) ∈ 𝕏 ⇒ ⋃_{n=1}^{∞} K_{n} = N(f);

• A ⊂ K_{↗}^{ℕ} is residual ⇒ for a typical f ∈ C(I), ∃(K_{n}) ∈ A ((K_{n}), f) ∈ 𝕏.

⟲⊞ 16(↱)

(1) ⇒ (2)

Suppose N(f) ∈ F for a typical f ∈ C(I). Take a dense G_{δ} set G ⊂ {f ∈ C(I) | N(f) ∈ F}. A := {(K_{n}) ∈ K_{↗}^{ℕ} | ∃f ∈ G ((K_{n}), f) ∈ 𝕏}.

(K_{n}) ∈ A implies ⋃_{n=1}^{∞} K_{n} = N(f) ∈ F. Thus it suffices to show that A is residual.

Thm (1) N(f) ∈ F for a typical f

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

Lem 𝕏 ⊂ K_{↗}^{ℕ} × C(I) satisfies

• ((K_{n}), f) ∈ 𝕏 ⇒ ⋃_{n=1}^{∞} K_{n} = N(f);

• A ⊂ K_{↗}^{ℕ} is residual ⇒ for a typical f ∈ C(I), ∃(K_{n}) ∈ A ((K_{n}), f) ∈ 𝕏.

⟲⊞ 17(↱)

A turns out to be analytic (since 𝕏 is ‘good’).

∴ A has the Baire Property.

∴ A is either meagre or residual (topological 0-1 law).

Suppose A is meagre.

Then A^{c} is residual.

Thm (1) N(f) ∈ F for a typical f

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

Lem 𝕏 ⊂ K_{↗}^{ℕ} × C(I) satisfies

• ((K_{n}), f) ∈ 𝕏 ⇒ ⋃_{n=1}^{∞} K_{n} = N(f);

• A ⊂ K_{↗}^{ℕ} is residual ⇒ for a typical f ∈ C(I), ∃(K_{n}) ∈ A ((K_{n}), f) ∈ 𝕏.

Is A = {(K_{n}) ∈ K_{↗}^{ℕ} | ∃f ∈ G ((K_{n}), f) ∈ 𝕏} residual?

⟲⊞ 18(↱)

By Lem, for a typical f ∈ C(I),

∃(K_{n}) ∈ A^{c} ((K_{n}), f) ∈ 𝕏.

Thus

∃f ∈ G ∃(K_{n}) ∈ A^{c} ((K_{n}), f) ∈ 𝕏.

This contradicts the definition of A.

Thm (1) N(f) ∈ F for a typical f

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

Lem 𝕏 ⊂ K_{↗}^{ℕ} × C(I) satisfies

• ((K_{n}), f) ∈ 𝕏 ⇒ ⋃_{n=1}^{∞} K_{n} = N(f);

• A ⊂ K_{↗}^{ℕ} is residual ⇒ for a typical f ∈ C(I), ∃(K_{n}) ∈ A ((K_{n}), f) ∈ 𝕏.

A^{c} = {(K_{n}) ∈ K_{↗}^{ℕ} | ∀f ∈ G ((K_{n}), f) ∉ 𝕏} is assumed to be residual.

⟲⊞ 19(↱)

⟲  END  Bye  ⊞