# Continuous real distribution function

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Here a continuous distribution means that its CDF is continuous; the CDF must still be nondecreasing, right-continuous and have endpoint limits0 and1.

mko-continuous-real-distribution-function · definition · v0.10.0



## Mathematical statement

A continuous CDF has no jumps, so every singleton has probability0. 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.





## Representative formula (exact TeX)

F(x)





## Assumptions


- Exact positive families use linear interpolation between knots, constant0 on the left and constant1 on the right.
- Knot values are nondecreasing from0 to1, knots are strictly increasing, and audit query points must also be strictly ordered as exact rationals.





## Symbol scope


- F — continuous cumulative distribution function
- x — real threshold





## Intuition, definition, and misconceptions

CDF continuity is a property of the distribution.

Continuity implies zero point masses.

Continuity or atomlessness does not guarantee a density.

Do not identify a continuous CDF with existence of a PDF or demand strict increase.





## Adopted derivation guide

[A continuous CDF with a flat interval](https://ocme.evemisslab.com/en/derivations/derivation-mko-continuous-real-distribution-function-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Cumulative distribution function](https://ocme.evemisslab.com/en/mko/mko-cumulative-distribution-function/) — Retain every CDF obligation.
- [Point values and continuity of real functions](https://ocme.evemisslab.com/en/mko/mko-continuity/) — The CDF has no jumps.
- [Function limits on punctured real domains](https://ocme.evemisslab.com/en/mko/mko-function-limit/) — The endpoint limits are required.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — The CDF is nondecreasing.
- [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/mko/mko-sigma-algebra-measure-space/) — The distribution is a probability law.





## Verification boundary

Piecewise-linear families are exact instances; the singular boundary performs no density computation.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add the standard normal distribution.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/continuous-real-distribution-function.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-continuous-real-distribution-function/index.md)