Question 15747: Distinguish Required Continuity Points from Limit-CDF Jumps

Original question and givens

Use the explicit Bernoulli symbolic family with interior limit p=22/30, 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=9/7. 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=22/30, 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=9/7. 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

Step-by-step derivation

interior Bernoulli limit mass at0 := sub(1, 11/15)

interior Bernoulli limit mass at0 = 4/15

original limit parameter=[22, 30];reduced p=11/15;mass at0=4/15;mass at1=11/15

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

interior limit p=11/15;exact threshold=9/7;jump set=[0, 1]

limit CDF continuity point=Yes;definition obligation=CDF_convergence_required_here

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

(11/15, 9/7, 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.