Continuous real distribution function

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

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 — Retain every CDF obligation.
• Point values and continuity of real functions — The CDF has no jumps.
• Function limits on punctured real domains — The endpoint limits are required.
• Usual order on the real numbers — The CDF is nondecreasing.
• Sigma-algebras and measure spaces — 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.