# Question 22838: A Continuous CDF May Have Flat Intervals [Return to catalogue page 115](https://ocme.evemisslab.com/en/questions/page/115/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations/#q-20261014-advance-20261111-r04-continuous-real-distribution-function-0013) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0044-22838 · q-20261014-advance-20261111-r04-continuous-real-distribution-function-0013 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-continuous-real-distribution-function](https://ocme.evemisslab.com/en/mko/mko-continuous-real-distribution-function/) ## Step-by-step derivation S001 · impliesBefore source operation=contcdf-piecewise-linear-qualification;original snapshot and mathematics retained After explicit ordered source output fields=[knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate] Conditions: none Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore CDF segment0 positive knot width := sub(0, -14) After CDF segment0 positive knot width = 14 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore CDF segment0 nonnegative value increment := sub(13/14, 0) After CDF segment0 nonnegative value increment = 13/14 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore CDF segment0 exact nonnegative slope := div(13/14, 14) After CDF segment0 exact nonnegative slope = 13/196 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore CDF segment1 positive knot width := sub(14, 0) After CDF segment1 positive knot width = 14 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore CDF segment1 nonnegative value increment := sub(1, 13/14) After CDF segment1 nonnegative value increment = 1/14 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore CDF segment1 exact nonnegative slope := div(1/14, 14) After CDF segment1 exact nonnegative slope = 1/196 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore ORIGINAL strictly increasing knots=[-14, 0, 14];aligned exact nondecreasing values=[0, 13/14, 1];ALL [width,height,slope] rows=[[14, 13/14, 13/196], [14, 1/14, 1/196]] After declare linear interpolation on EVERY segment,constant0 left and1 right;shared knot values make joins continuous;all slopes>=0 allow flats;continuous implies right-continuous and endpoints give limits0/1 Conditions: none These are properties of the declared whole interpolation law,not inferences from finite knot/query observations.Nonnegative slopes and matching joins give monotonicity and continuity;constant tails give the limits.This PL example may have a density without every continuous CDF having one. Step identifiers and dependenciesRule: contcdf_WHOLE_piecewise_law_continuous_flat_monotone_tails · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute CDF segment1 exact nonnegative slope with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute CDF segment1 nonnegative value increment with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute CDF segment1 positive knot width with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute CDF segment0 exact nonnegative slope with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute CDF segment0 nonnegative value increment with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute CDF segment0 positive knot width with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([-14, 0, 14], [0, 13/14, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: dbda86f0c3276378199128102fe8ce5f9438d934f6d8b6651efbb1c03361f41a - candidate record SHA-256: a8902fdea0a8093e2b6053deaffeefcef08767dd2bc80648c41a457b3321bdc7 - semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81 - question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r04-continuous-real-distribution-function-0013.json)