Question 22897: The Singleton Mass Is the Jump, Not the CDF Value

Original question and givens

Use strictly increasing knots=[-23, 0, 23], aligned exact values=[0, 22/23, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).

• G001 — Use strictly increasing knots=[-23, 0, 23], aligned exact values=[0, 22/23, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).

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-continuous-real-distribution-function

Step-by-step derivation

source operation=contcdf-zero-jump-atomlessness;original snapshot and mathematics retained

explicit ordered source output fields=[point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate]

CDF segment0 positive knot width := sub(0, -23)

CDF segment0 positive knot width = 23

CDF segment0 nonnegative value increment := sub(22/23, 0)

CDF segment0 nonnegative value increment = 22/23

CDF segment0 exact nonnegative slope := div(22/23, 23)

CDF segment0 exact nonnegative slope = 22/529

CDF segment1 positive knot width := sub(23, 0)

CDF segment1 positive knot width = 23

CDF segment1 nonnegative value increment := sub(1, 22/23)

CDF segment1 nonnegative value increment = 1/23

CDF segment1 exact nonnegative slope := div(1/23, 23)

CDF segment1 exact nonnegative slope = 1/529

ORIGINAL strictly increasing knots=[-23, 0, 23];aligned exact nondecreasing values=[0, 22/23, 1];ALL [width,height,slope] rows=[[23, 22/23, 22/529], [23, 1/23, 1/529]]

declare linear interpolation on EVERY segment,constant0 left and1 right;shared knot values make joins continuous;all slopes>=0 allow flats;continuous implies right-continuous and endpoints give limits0/1

CDF ORIGINAL atom query ORIGINAL query minus left knot := sub(0, -23)

CDF ORIGINAL atom query ORIGINAL query minus left knot = 23

CDF ORIGINAL atom query exact interval fraction := div(23, 23)

CDF ORIGINAL atom query exact interval fraction = 1

CDF ORIGINAL atom query fraction times ORIGINAL value increment := mul(1, 22/23)

CDF ORIGINAL atom query fraction times ORIGINAL value increment = 22/23

CDF ORIGINAL atom query plus ORIGINAL left-knot value := sum(0, 22/23)

CDF ORIGINAL atom query plus ORIGINAL left-knot value = 22/23

CDF ORIGINAL atom query:ORIGINAL exact query x=0;knots=[-23, 0, 23];values=[0, 22/23, 1]

case=FIRST source interval containing query,including shared knot;source interval index=0;F(x)=22/23;this is CDF value,NOT singleton mass

CDF actual value minus equal continuous left limit := sub(22/23, 22/23)

CDF actual value minus equal continuous left limit = 0

ORIGINAL point=0;WHOLE-law continuity gives F(x)=22/23=F(x-)

P(X=x)=F(x)-F(x-)=0;zero_jump=Yes;F(x) may be positive;atomless alone does NOT imply a density

Reverse checks

• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute CDF actual value minus equal continuous left limit with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute CDF ORIGINAL atom query plus ORIGINAL left-knot value with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute CDF ORIGINAL atom query fraction times ORIGINAL value increment with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute CDF ORIGINAL atom query exact interval fraction with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute CDF ORIGINAL atom query ORIGINAL query minus left knot with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute CDF segment1 exact nonnegative slope with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute CDF segment1 nonnegative value increment with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute CDF segment1 positive knot width with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute CDF segment0 exact nonnegative slope with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute CDF segment0 nonnegative value increment with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute CDF segment0 positive knot width with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(0, 22/23, 22/23, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)

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.