# Question 15730: Distinguish Required Continuity Points from Limit-CDF Jumps

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0030-15730 · q-20261014-advance-20261031-r03-convergence-in-distribution-0055



## Original question and givens

Use the explicit Bernoulli symbolic family with interior limit p=5/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-8/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.


- G001 — Use the explicit Bernoulli symbolic family with interior limit p=5/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-8/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-convergence-in-distribution](https://ocme.evemisslab.com/en/mko/mko-convergence-in-distribution/)





## Step-by-step derivation

S001 · calculationBefore

interior Bernoulli limit mass at0 := sub(1, 1/6)

After

interior Bernoulli limit mass at0 = 5/6

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · impliesBefore

original limit parameter=[5, 30];reduced p=1/6;mass at0=5/6;mass at1=1/6

After

both endpoint masses positive;only limit-CDF jumps at0 and1;registered interior projection

Conditions: none

Honor the source English restriction0<p<1,so both jumps genuinely have positive mass.Although the raw oracle allows endpoint parameters,do not extend this continuity-point protocol to degenerate distributions;their jump sets require separate analysis.

Step identifiers and dependenciesRule: Bernoulli_interior_English_guard_not_degenerate_jump_set · Uses: G001, S001





S003 · impliesBefore

interior limit p=1/6;exact threshold=-8/7;jump set=[0, 1]

After

limit CDF continuity point=Yes;definition obligation=CDF_convergence_required_here

Conditions: none

Convergence in distribution requires convergence at every continuity point of the limit CDF.A jump point is not required by the definition;this is not a claim of failure there,and continuity is not decided from one n.

Step identifiers and dependenciesRule: convdist_limit_continuity_obligation_not_convergence_failure · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Recompute interior Bernoulli limit mass at0 with exact rational arithmetic, not rounded decimals.





## Result

(1/6, -8/7, Yes, CDF_convergence_required_here)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 0c7f923d72a6c40a15962cd633e34640bf722c3bbeb0b8203992f63b704ba1b9
- candidate record SHA-256: d5e3fd9bfd304a76fddf5314e0be2671b0a55b18f86f376bf470c0ab057d408b
- semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc
- question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r03-convergence-in-distribution-0055.json)