A continuous CDF with a flat interval

Goal and scope

Construct a complete knot-based CDF and audit atomlessness.

An exact piecewise-linear family; continuity alone will not imply general density existence.

Givens and symbols

• work

• G1 — The knots are(0,0),(1,1/2),(2,1/2),(3,1); interpolate linearly, with left tail 0 and right tail 1.

• F — real CDF
• t — arbitrary real threshold

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• mko_definition · cited — A continuous CDF has no jumps, so every singleton has probability 0. This is not continuity of a random-variable map on the sample space and does not imply a density or absolute continuity. An affine Cantor distribution supplies a continuous, atomless, singular boundary. Flat CDF intervals are allowed; strict increase is not required. · mko-continuous-real-distribution-function
• work · prerequisite — Piecewise-linear functions are continuous on each piece; matching junctions ensure global continuity; CDF existence and the jump identity are prerequisites.

Main derivation

0<1<2<3,\quad0\le1/2\le1/2\le1

\text{strict knot order and nondecreasing levels}

t\le0\text{ or }t\ge3

F(t)=0\ (t\le0),\quad F(t)=1\ (t\ge3)

0\le t\le1

F(t)=t/2

1\le t\le2

F(t)=1/2

2\le t\le3

F(t)=1/2+(t-2)/2=(t-1)/2

t\in\{0,1,2,3\}

F(t^-)=F(t)=F(t^+)

F\text{ nondecreasing and continuous, with tails }0,1

\exists X:\ F(t)=P(X\le t);\quad P(X=t)=F(t)-F(t^-)=0\quad(t\in\mathbb R)

\lim_{t\to-\infty}F(t)=0,\quad\lim_{t\to\infty}F(t)=1

F\text{ is a continuous CDF, not strictly increasing}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — General continuous CDFs may be singular, as with a Cantor law; this example cannot imply densities generally.
• S07 · premise_audit · G1, S06, work — Obtain the law before writing P(X=t); atomless does not mean zero interval mass, since P(0<X≤1)=1/2.
• S06 · premise_audit · G1, S05, work — Auditing every junction proves global continuity, not sampling.
• S05 · premise_audit · G1, S04, work — At t=2 it must give 1/2, not the incorrect continuation t/2.
• S04 · premise_audit · G1, S03, work — A CDF need not be strictly increasing.
• S03 · premise_audit · G1, S02, work — Substitution at 0 and 1 recovers both knots.
• S02 · premise_audit · G1, S01, work — The values at endpoints t=0 and t=3 must agree with the interpolated pieces.
• S01 · premise_audit · G1, work — Equal levels are allowed; only x-coordinates must be strictly ordered.

Error controls and negative controls

Junctions

Check both sides at 0,1,2,3.

Query ordering

Queries1/2,3/2,5/2 are strictly ordered and yield1/4,1/2,3/4.

• Every continuous CDF is strictly increasing. → reject — This example is constant 1/2 on[1,2].
• A continuous CDF must have a density. → reject — A singular Cantor-type law is a counterexample.
• P(X=1)=F(1)=1/2. → reject — Singleton mass is the jump, which is zero here.

Result and limitations

This is a continuous atomless CDF with flat interval[1,2].

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 79d7922fc70bd68789e0c257bdf10bf05aae23f17410f0343d79c5d7cb7756de
• original source SHA-256: 5f3d06982951d8148e60031cd501f7fc189242eca7f7a442857ec8ad6f770380
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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