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Under Dependent Choice, nowhere differentiable functions form a residual subset of $C ([0, 1], R)$

Statement

Assume Dependent Choice. The nowhere differentiable functions form a residual subset of $C ([0, 1], R)$ with the uniform metric, where differentiability at an endpoint means the corresponding one-sided derivative.

Facts & Assumptions

Given: The objects, hypotheses, and choice principles stated above.

[F1]

Let $X$ be a topological space and let $A \subseteq X$ . The set $A$ is nowhere dense when $int (\overline{A}) = \emptyset$ (def-interior-closure-boundary-top). It is meagre when there is a sequence $(N_{n})_{n \in N}$ of nowhere dense subsets of $X$ with $A \subseteq ⋃_{n} N_{n}$ . It is residual, or comeagre, when $X ∖ A$ is meagre. The empty union shows that $\emptyset$ is meagre, including when $X = \emptyset$ . (Nowhere dense, meagre, residual, and comeagre subsets of a topological space).

[F2]

For $p, q \in N_{> 0}$ , let $E_{p, q}$ be the functions $f \in C ([0, 1], R)$ for which some $a \in [0, 1]$ satisfies $∣ f (t) - f (a) ∣ \leq p ∣ t - a ∣$ whenever $t \in [0, 1]$ and $∣ t - a ∣ < 1 / q$ . Then $E_{p, q}$ is closed in the supremum metric. (Functions satisfying a fixed local Lipschitz bound somewhere form a closed subset of $C ([0, 1])$ ).

[F3]

For every $f \in C ([0, 1], R)$ , every $ε > 0$ , and every $M > 0$ , there is a piecewise-affine $h$ with finitely many vertices such that $∥ f - h ∥_{\infty} < ε$ and every slope on a nonvertex affine piece has absolute value greater than $M$ . (Polygonal functions with sufficiently steep nonvertex slopes are dense in $C ([0, 1])$ ).

[F4]

Assume the Axiom of Dependent Choice ( $D C$ ). Then the set of continuous functions $[0, 1] \to R$ having no finite two-sided derivative at an interior point and no finite one-sided derivative at either endpoint is dense in $C ([0, 1], R)$ for the supremum metric. (Under Dependent Choice, continuous nowhere differentiable functions form a dense subset of $C ([0, 1], R)$ ).

[F5]

For every topological space $X$ , the meagre subsets of $X$ contain $\emptyset$ , are closed under taking subsets, and are closed under countable unions (The meagre subsets of a topological space form a sigma-ideal).

Proof

technique · direct

1.1

Use the published closed pointwise-Lipschitz sets and the dense steep-polygonal perturbations to show every such closed set has empty interior.

givenF3F1F2

2.1

Their countable union $M$ is meagre by step 1.1 and contains every function with a finite derivative at some point, so the complement of $M$ is residual and consists of nowhere differentiable functions. The set $N$ of nowhere differentiable functions contains that residual complement, so its own complement is a subset of $M$ ; meagre sets are closed downward under subsets, being a sigma-ideal [F5], hence the complement of $N$ is meagre and $N$ is residual. One-sided endpoint derivatives are included.

step 1.1F4F1F3F5

3.1

The preceding construction and implications establish the assertion.

step 2.1∎

Depends on

Used by

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Sources

David Marker, Descriptive Set Theory, §§1–2 (standard reference, not scraped)
Michael Kunzinger, General Topology, §§11.3–11.4 (standard reference, not scraped)
MFF General Topology course summary, §4.3 (standard reference, not scraped)

Under Dependent Choice, nowhere differentiable functions form a residual subset of C([0,1],R)C([0,1],R)