Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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≥1n\ge1, the interval endpoints an=1/(n+1)a_n=1/(n+1), bn=1/nb_n=1/n, midpoint cn=(an+bn)/2c_n=(a_n+b_n)/2, and radius rn=(bn−an)/2r_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 fn(x)=max⁡{0,1−∣x−cn∣/rn}f_n(x)=\max\{0,1-|x-c_n|/r_n\}. Each fnf_n is continuous by [L1], is supported in [an,bn][a_n,b_n], and satisfies fn(cn)=1f_n(c_n)=1.

L1construct
2.1

The cozero sets (an,bn)(a_n,b_n) are pairwise disjoint, so f=∑n≥1fnf=\sum_{n\ge1}f_n is pointwise finite and f(0)=0f(0)=0.

step 1.1
3.1

Since cn→0c_n\to0 while f(cn)=1f(c_n)=1 for every nn, ff is not continuous at 00.

step 1.1step 2.1
4.1

The cozero family is not locally finite at 00, 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 44 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources