Whole Continuous CDFs, Zero Jumps, Flat Intervals and Singular Boundaries
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-2, 0, 2], aligned exact values=[0, 1/2, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-2, 0, 2], [0, 1/2, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-2, 0, 2], [0, 1/2, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
2. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-3, 0, 3], aligned exact values=[0, 2/3, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-3, 0, 3], [0, 2/3, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-3, 0, 3], [0, 2/3, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
3. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-4, 0, 4], aligned exact values=[0, 3/4, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-4, 0, 4], [0, 3/4, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-4, 0, 4], [0, 3/4, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
4. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-5, 0, 5], aligned exact values=[0, 4/5, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-5, 0, 5], [0, 4/5, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-5, 0, 5], [0, 4/5, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
5. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-6, 0, 6], aligned exact values=[0, 5/6, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-6, 0, 6], [0, 5/6, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-6, 0, 6], [0, 5/6, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
6. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-7, 0, 7], aligned exact values=[0, 6/7, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-7, 0, 7], [0, 6/7, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-7, 0, 7], [0, 6/7, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
7. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-8, 0, 8], aligned exact values=[0, 7/8, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-8, 0, 8], [0, 7/8, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-8, 0, 8], [0, 7/8, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
8. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-9, 0, 9], aligned exact values=[0, 8/9, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-9, 0, 9], [0, 8/9, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-9, 0, 9], [0, 8/9, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
9. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-10, 0, 10], aligned exact values=[0, 9/10, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-10, 0, 10], [0, 9/10, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-10, 0, 10], [0, 9/10, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
10. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-11, 0, 11], aligned exact values=[0, 10/11, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-11, 0, 11], [0, 10/11, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-11, 0, 11], [0, 10/11, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
11. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-12, 0, 12], aligned exact values=[0, 11/12, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-12, 0, 12], [0, 11/12, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-12, 0, 12], [0, 11/12, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
12. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-13, 0, 13], aligned exact values=[0, 12/13, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-13, 0, 13], [0, 12/13, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-13, 0, 13], [0, 12/13, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
13. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-14, 0, 14], aligned exact values=[0, 13/14, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-14, 0, 14], [0, 13/14, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-14, 0, 14], [0, 13/14, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
14. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-15, 0, 15], aligned exact values=[0, 14/15, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-15, 0, 15], [0, 14/15, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-15, 0, 15], [0, 14/15, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
15. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-16, 0, 16], aligned exact values=[0, 15/16, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-16, 0, 16], [0, 15/16, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-16, 0, 16], [0, 15/16, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
16. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-17, 0, 17], aligned exact values=[0, 16/17, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-17, 0, 17], [0, 16/17, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-17, 0, 17], [0, 16/17, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
17. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-18, 0, 18], aligned exact values=[0, 17/18, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-18, 0, 18], [0, 17/18, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-18, 0, 18], [0, 17/18, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
18. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-19, 0, 19], aligned exact values=[0, 18/19, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-19, 0, 19], [0, 18/19, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-19, 0, 19], [0, 18/19, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
19. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-20, 0, 20], aligned exact values=[0, 19/20, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-20, 0, 20], [0, 19/20, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-20, 0, 20], [0, 19/20, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
20. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-21, 0, 21], aligned exact values=[0, 20/21, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-21, 0, 21], [0, 20/21, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-21, 0, 21], [0, 20/21, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
21. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-22, 0, 22], aligned exact values=[0, 21/22, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-22, 0, 22], [0, 21/22, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-22, 0, 22], [0, 21/22, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
22. A Continuous CDF May Have Flat Intervals
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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-23, 0, 23], [0, 22/23, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-23, 0, 23], [0, 22/23, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
23. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-24, 0, 24], aligned exact values=[0, 23/24, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-24, 0, 24], [0, 23/24, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-24, 0, 24], [0, 23/24, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
24. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-25, 0, 25], aligned exact values=[0, 24/25, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-25, 0, 25], [0, 24/25, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-25, 0, 25], [0, 24/25, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
25. A Continuous CDF May Have Flat Intervals
Use strictly increasing knots=[-26, 0, 26], aligned exact values=[0, 25/26, 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. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).
View answer and explanation
([-26, 0, 26], [0, 25/26, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)
Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-26, 0, 26], [0, 25/26, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-piecewise-linear-qualification · candidate
26. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-2, 0, 2], aligned exact values=[0, 1/2, 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. Exact query_points=[-3, 0, 3] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-3, 0], [0, 1/2], [3, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-3, 0], [0, 1/2], [3, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
27. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-3, 0, 3], aligned exact values=[0, 2/3, 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. Exact query_points=[-4, 0, 4] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-4, 0], [0, 2/3], [4, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-4, 0], [0, 2/3], [4, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
28. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-4, 0, 4], aligned exact values=[0, 3/4, 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. Exact query_points=[-5, 0, 5] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-5, 0], [0, 3/4], [5, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-5, 0], [0, 3/4], [5, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
29. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-5, 0, 5], aligned exact values=[0, 4/5, 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. Exact query_points=[-6, 0, 6] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-6, 0], [0, 4/5], [6, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-6, 0], [0, 4/5], [6, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
30. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-6, 0, 6], aligned exact values=[0, 5/6, 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. Exact query_points=[-7, 0, 7] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-7, 0], [0, 5/6], [7, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-7, 0], [0, 5/6], [7, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
31. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-7, 0, 7], aligned exact values=[0, 6/7, 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. Exact query_points=[-8, 0, 8] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-8, 0], [0, 6/7], [8, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-8, 0], [0, 6/7], [8, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
32. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-8, 0, 8], aligned exact values=[0, 7/8, 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. Exact query_points=[-9, 0, 9] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-9, 0], [0, 7/8], [9, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-9, 0], [0, 7/8], [9, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
33. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-9, 0, 9], aligned exact values=[0, 8/9, 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. Exact query_points=[-10, 0, 10] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-10, 0], [0, 8/9], [10, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-10, 0], [0, 8/9], [10, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
34. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-10, 0, 10], aligned exact values=[0, 9/10, 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. Exact query_points=[-11, 0, 11] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-11, 0], [0, 9/10], [11, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-11, 0], [0, 9/10], [11, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
35. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-11, 0, 11], aligned exact values=[0, 10/11, 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. Exact query_points=[-12, 0, 12] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-12, 0], [0, 10/11], [12, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-12, 0], [0, 10/11], [12, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
36. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-12, 0, 12], aligned exact values=[0, 11/12, 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. Exact query_points=[-13, 0, 13] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-13, 0], [0, 11/12], [13, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-13, 0], [0, 11/12], [13, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
37. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-13, 0, 13], aligned exact values=[0, 12/13, 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. Exact query_points=[-14, 0, 14] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-14, 0], [0, 12/13], [14, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-14, 0], [0, 12/13], [14, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
38. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-14, 0, 14], aligned exact values=[0, 13/14, 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. Exact query_points=[-15, 0, 15] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-15, 0], [0, 13/14], [15, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-15, 0], [0, 13/14], [15, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
39. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-15, 0, 15], aligned exact values=[0, 14/15, 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. Exact query_points=[-16, 0, 16] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-16, 0], [0, 14/15], [16, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-16, 0], [0, 14/15], [16, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
40. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-16, 0, 16], aligned exact values=[0, 15/16, 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. Exact query_points=[-17, 0, 17] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-17, 0], [0, 15/16], [17, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-17, 0], [0, 15/16], [17, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
41. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-17, 0, 17], aligned exact values=[0, 16/17, 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. Exact query_points=[-18, 0, 18] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-18, 0], [0, 16/17], [18, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-18, 0], [0, 16/17], [18, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
42. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-18, 0, 18], aligned exact values=[0, 17/18, 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. Exact query_points=[-19, 0, 19] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-19, 0], [0, 17/18], [19, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-19, 0], [0, 17/18], [19, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
43. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-19, 0, 19], aligned exact values=[0, 18/19, 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. Exact query_points=[-20, 0, 20] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-20, 0], [0, 18/19], [20, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-20, 0], [0, 18/19], [20, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
44. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-20, 0, 20], aligned exact values=[0, 19/20, 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. Exact query_points=[-21, 0, 21] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-21, 0], [0, 19/20], [21, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-21, 0], [0, 19/20], [21, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
45. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-21, 0, 21], aligned exact values=[0, 20/21, 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. Exact query_points=[-22, 0, 22] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-22, 0], [0, 20/21], [22, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-22, 0], [0, 20/21], [22, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
46. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-22, 0, 22], aligned exact values=[0, 21/22, 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. Exact query_points=[-23, 0, 23] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-23, 0], [0, 21/22], [23, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-23, 0], [0, 21/22], [23, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
47. Strictly Ordered Query Points May Have Equal CDF Values
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. Exact query_points=[-24, 0, 24] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-24, 0], [0, 22/23], [24, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-24, 0], [0, 22/23], [24, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
48. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-24, 0, 24], aligned exact values=[0, 23/24, 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. Exact query_points=[-25, 0, 25] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-25, 0], [0, 23/24], [25, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-25, 0], [0, 23/24], [25, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
49. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-25, 0, 25], aligned exact values=[0, 24/25, 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. Exact query_points=[-26, 0, 26] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-26, 0], [0, 24/25], [26, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-26, 0], [0, 24/25], [26, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
50. Strictly Ordered Query Points May Have Equal CDF Values
Use strictly increasing knots=[-26, 0, 26], aligned exact values=[0, 25/26, 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. Exact query_points=[-27, 0, 27] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).
View answer and explanation
([[-27, 0], [0, 25/26], [27, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)
Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-27, 0], [0, 25/26], [27, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-monotone-endpoint-audit · candidate
51. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-2, 0, 2], aligned exact values=[0, 1/2, 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).
View answer and explanation
(0, 1/2, 1/2, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 1/2, 1/2, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
52. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-3, 0, 3], aligned exact values=[0, 2/3, 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).
View answer and explanation
(0, 2/3, 2/3, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 2/3, 2/3, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
53. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-4, 0, 4], aligned exact values=[0, 3/4, 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).
View answer and explanation
(0, 3/4, 3/4, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 3/4, 3/4, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
54. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-5, 0, 5], aligned exact values=[0, 4/5, 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).
View answer and explanation
(0, 4/5, 4/5, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 4/5, 4/5, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
55. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-6, 0, 6], aligned exact values=[0, 5/6, 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).
View answer and explanation
(0, 5/6, 5/6, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 5/6, 5/6, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
56. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-7, 0, 7], aligned exact values=[0, 6/7, 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).
View answer and explanation
(0, 6/7, 6/7, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 6/7, 6/7, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
57. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-8, 0, 8], aligned exact values=[0, 7/8, 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).
View answer and explanation
(0, 7/8, 7/8, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 7/8, 7/8, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
58. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-9, 0, 9], aligned exact values=[0, 8/9, 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).
View answer and explanation
(0, 8/9, 8/9, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 8/9, 8/9, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
59. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-10, 0, 10], aligned exact values=[0, 9/10, 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).
View answer and explanation
(0, 9/10, 9/10, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 9/10, 9/10, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
60. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-11, 0, 11], aligned exact values=[0, 10/11, 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).
View answer and explanation
(0, 10/11, 10/11, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 10/11, 10/11, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
61. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-12, 0, 12], aligned exact values=[0, 11/12, 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).
View answer and explanation
(0, 11/12, 11/12, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 11/12, 11/12, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
62. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-13, 0, 13], aligned exact values=[0, 12/13, 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).
View answer and explanation
(0, 12/13, 12/13, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 12/13, 12/13, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
63. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-14, 0, 14], aligned exact values=[0, 13/14, 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).
View answer and explanation
(0, 13/14, 13/14, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 13/14, 13/14, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
64. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-15, 0, 15], aligned exact values=[0, 14/15, 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).
View answer and explanation
(0, 14/15, 14/15, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 14/15, 14/15, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
65. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-16, 0, 16], aligned exact values=[0, 15/16, 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).
View answer and explanation
(0, 15/16, 15/16, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 15/16, 15/16, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
66. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-17, 0, 17], aligned exact values=[0, 16/17, 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).
View answer and explanation
(0, 16/17, 16/17, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 16/17, 16/17, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
67. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-18, 0, 18], aligned exact values=[0, 17/18, 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).
View answer and explanation
(0, 17/18, 17/18, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 17/18, 17/18, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
68. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-19, 0, 19], aligned exact values=[0, 18/19, 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).
View answer and explanation
(0, 18/19, 18/19, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 18/19, 18/19, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
69. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-20, 0, 20], aligned exact values=[0, 19/20, 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).
View answer and explanation
(0, 19/20, 19/20, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 19/20, 19/20, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
70. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-21, 0, 21], aligned exact values=[0, 20/21, 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).
View answer and explanation
(0, 20/21, 20/21, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 20/21, 20/21, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
71. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-22, 0, 22], aligned exact values=[0, 21/22, 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).
View answer and explanation
(0, 21/22, 21/22, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 21/22, 21/22, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
72. The Singleton Mass Is the Jump, Not the CDF Value
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).
View answer and explanation
(0, 22/23, 22/23, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 22/23, 22/23, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
73. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-24, 0, 24], aligned exact values=[0, 23/24, 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).
View answer and explanation
(0, 23/24, 23/24, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 23/24, 23/24, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
74. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-25, 0, 25], aligned exact values=[0, 24/25, 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).
View answer and explanation
(0, 24/25, 24/25, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 24/25, 24/25, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
75. The Singleton Mass Is the Jump, Not the CDF Value
Use strictly increasing knots=[-26, 0, 26], aligned exact values=[0, 25/26, 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).
View answer and explanation
(0, 25/26, 25/26, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)
Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 25/26, 25/26, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-zero-jump-atomlessness · candidate
76. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-1,2], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-1, 2, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-1, 2, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
77. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-2,3], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-2, 3, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-2, 3, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
78. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-3,4], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-3, 4, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-3, 4, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
79. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-4,5], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-4, 5, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-4, 5, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
80. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-5,6], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-5, 6, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-5, 6, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
81. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-6,7], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-6, 7, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-6, 7, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
82. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-7,8], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-7, 8, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-7, 8, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
83. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-8,9], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-8, 9, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-8, 9, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
84. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-9,10], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-9, 10, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-9, 10, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
85. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-10,11], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-10, 11, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-10, 11, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
86. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-11,12], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-11, 12, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-11, 12, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
87. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-12,13], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-12, 13, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-12, 13, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
88. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-13,14], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-13, 14, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-13, 14, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
89. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-14,15], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-14, 15, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-14, 15, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
90. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-15,16], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-15, 16, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-15, 16, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
91. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-16,17], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-16, 17, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-16, 17, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
92. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-17,18], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-17, 18, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-17, 18, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
93. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-18,19], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-18, 19, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-18, 19, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
94. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-19,20], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-19, 20, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-19, 20, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
95. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-20,21], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-20, 21, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-20, 21, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
96. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-21,22], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-21, 22, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-21, 22, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
97. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-22,23], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-22, 23, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-22, 23, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
98. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-23,24], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-23, 24, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-23, 24, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
99. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-24,25], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-24, 25, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-24, 25, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
100. Continuous and Atomless Does Not Imply a Density
Use the standard Cantor distribution affinely carried onto [-25,26], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples.
View answer and explanation
(-25, 26, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-25, 26, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density).
Reviewed step-by-step derivation
Continuous real distribution function · qf-mod42-contcdf-singular-no-density-boundary · candidate
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