CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Without local finiteness, a pointwise finite sum of continuous functions can be discontinuous

Statement refuted

Every pointwise finite sum of continuous real-valued functions is continuous.

Facts & Assumptions

Given: For $n \geq 1$ , the interval endpoints $a_{n} = 1 / (n + 1)$ , $b_{n} = 1 / n$ , midpoint $c_{n} = (a_{n} + b_{n}) / 2$ , and radius $r_{n} = (b_{n} - a_{n}) / 2$ .

[L1]

Maxima, absolute values, and finite algebraic combinations of continuous real functions are continuous (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined).

[L2]

A locally finite family has a continuous pointwise sum (A locally finite family of continuous nonnegative functions has a continuous pointwise sum).

Counterexample

technique · direct

1.1

Define $f_{n} (x) = \max {0, 1 - ∣ x - c_{n} ∣ / r_{n}}$ . Each $f_{n}$ is continuous by [L1], is supported in $[a_{n}, b_{n}]$ , and satisfies $f_{n} (c_{n}) = 1$ .

L1construct

2.1

The cozero sets $(a_{n}, b_{n})$ are pairwise disjoint, so $f = \sum_{n \geq 1} f_{n}$ is pointwise finite and $f (0) = 0$ .

step 1.1

3.1

Since $c_{n} \to 0$ while $f (c_{n}) = 1$ for every $n$ , $f$ is not continuous at $0$ .

step 1.1step 2.1

4.1

The cozero family is not locally finite at $0$ , so this example does not contradict [L2] and refutes the displayed pointwise-finite claim.

L2step 2.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

J. Robbin, Partitions of Unity (standard reference, not scraped)