# Question 22863: Continuous and Atomless Does Not Imply a Density

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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.

R0044-22863 · q-20261014-advance-20261111-r04-continuous-real-distribution-function-0088



## Original question and givens

Use the standard Cantor distribution affinely carried onto [-13,14], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).


- G001 — Use the standard Cantor distribution affinely carried onto [-13,14], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).





## 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](https://ocme.evemisslab.com/en/mko/mko-continuous-real-distribution-function/)





## Step-by-step derivation

S001 · impliesBefore

source operation=contcdf-singular-no-density-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[left, right, CDF_continuous, atomless, has_density, singular, boundary]

Conditions: none

Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · calculationBefore

Cantor affine support positive width := sub(14, -13)

After

Cantor affine support positive width = 27

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

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





S003 · impliesBefore

ORIGINAL affine Cantor support[-13,14];positive width W=27

After

take independent fair digits B_k in{0,1},Y=sum_(k>=1)2*B_k/3^k and X=L+W*Y;the geometric tail converges and X lies in the affine Cantor support

Conditions: none

This defines the whole Cantor law,not the piecewise-linear constructor or a finite CDF sample.The convergent ternary0/2 series has probability1 on Cantor support,and the positive affine transformation is nondegenerate.

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





S004 · impliesBefore

ORIGINAL support width W=27;level n ternary construction uses2^n intervals of length W/3^n

After

total level-n length=W*(2/3)^n tends0,so support has Lebesgue measure0;any singleton has probability<=2*2^(-n) for EVERY n,hence atom probability0;CDF jumps are0

Conditions: none

The nested level sets contain the support and their total length tends0,proving Lebesgue-null support.A fixed point is bounded by at most two relevant cylinder masses2^(-n).The bound for every n proves no atoms,not sampling finitely many points.

Step identifiers and dependenciesRule: contcdf_Cantor_null_support_and_vanishing_atom_bounds · Uses: G001, S003





S005 · impliesBefore

ORIGINAL affine law on[-13,14] has support N with Lebesgue measure0 and probability1;all singleton probabilities0

After

CDF_continuous=Yes;atomless=Yes;has_density=No;singular=Yes;any Lebesgue density would integrate to0 over N,contradicting probability1

Conditions: none

CDF jumps equal singleton masses;an everywhere atomless law has a continuous CDF.The full-probability null support proves singularity and rules out a Lebesgue density,since its integral over that support would be0.This is a particular counterexample to continuity/atomlessness implying density.

Step identifiers and dependenciesRule: contcdf_Cantor_continuous_atomless_singular_NO_density · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Recompute Cantor affine support positive 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

(-13, 14, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 232bf805d8f3e535556b90e04dfa1b8f82327146c3ad343984072db1d8a6b360
- candidate record SHA-256: 5021598fdfea3d6503b005ddb58f9bd351d5afe3e14bb54f16c9101482a0c611
- semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
- question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r04-continuous-real-distribution-function-0088.json)