Question 22827: A Continuous CDF May Have Flat Intervals

Original question and givens

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).

• G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-continuous-real-distribution-function

Step-by-step derivation

source operation=contcdf-piecewise-linear-qualification;original snapshot and mathematics retained

explicit ordered source output fields=[knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate]

CDF segment0 positive knot width := sub(0, -3)

CDF segment0 positive knot width = 3

CDF segment0 nonnegative value increment := sub(2/3, 0)

CDF segment0 nonnegative value increment = 2/3

CDF segment0 exact nonnegative slope := div(2/3, 3)

CDF segment0 exact nonnegative slope = 2/9

CDF segment1 positive knot width := sub(3, 0)

CDF segment1 positive knot width = 3

CDF segment1 nonnegative value increment := sub(1, 2/3)

CDF segment1 nonnegative value increment = 1/3

CDF segment1 exact nonnegative slope := div(1/3, 3)

CDF segment1 exact nonnegative slope = 1/9

ORIGINAL strictly increasing knots=[-3, 0, 3];aligned exact nondecreasing values=[0, 2/3, 1];ALL [width,height,slope] rows=[[3, 2/3, 2/9], [3, 1/3, 1/9]]

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

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

([-3, 0, 3], [0, 2/3, 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.