False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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FALSE: a pointwise limit of continuous functions is continuous almost everywhere

Statement

False claim. A pointwise limit of continuous functions on $R$ is continuous almost everywhere.

Facts & Assumptions

Given: The fat Cantor set $S \subseteq [0, 1]$ .

[L1]

The fat Cantor set is closed, nowhere dense, and not Lebesgue null. (The Smith-Volterra-Cantor set is compact, perfect and nowhere dense, and does not have measure zero)

[L2]

Every closed subset of a metric space is the zero set of a continuous real function. (In a metric space every closed set is a zero set and a $G_{δ}$ , and the distance function separates a point from a closed set, so every metrizable space is Tychonoff and perfectly normal)

Refutation

technique · direct

1.1

By [L2], there is a continuous function $u : R \to R$ with [L2, choose] $S = {x : u (x) = 0}$ . Put $f_{0} : = f_{1}$ , and for $n \geq 1$ define

$f_{n} (x) : = \frac{1}{1 + n u (x)^{2}} .$

Each $f_{n}$ is continuous, because it is built from $u$ by continuous algebraic operations and the denominator is everywhere positive. [L2, choose]

2.1

If $x \in S$ , then $u (x) = 0$ and $f_{n} (x) = 1$ for every $n$ . If $x \notin S$ , [step 1.1, L1] then $u (x)^{2} > 0$ , so $1 + n u (x)^{2} \to + \infty$ and $f_{n} (x) \to 0$ . Thus the pointwise limit is $1_{S}$ . Since $S$ is closed and has empty interior by [L1], every point of $S$ is a boundary point of $S$ , and the indicator $1_{S}$ is discontinuous at every such point. Because $S$ is not Lebesgue null by [L1], the discontinuity set has positive measure.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

59 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Smith-Volterra-Cantor set (Wikipedia) (standard reference, not scraped)