A function attributed to Conway, made exact

Every Interval, the Whole Line

Type a decimal interval and a target, then build an exact rational point inside that interval which Conway's base-13 construction sends to the target. A second live instrument exposes the dense, measure-zero fractal where the function can be nonzero.

The graph cannot be honestly sampled. Almost every input goes to zero, yet every open interval contains an input for every real output. So this page does the one thing a plot cannot: it manufactures a witness and checks every digit.

Layer one: put a value inside an interval

Name the room. Name the value.

Use terminating decimals. The constructor finds a complete base- cylinder inside the interval, appends the decimal target as control digits, and returns the resulting rational exactly.

Exact witness constructor ready

Constructing the default witness...

A = positive output, B = negative output, C = decimal point
prefix · B 3 C 1 4 0 0 0 ... decodes as -3.14000...

The base- point is read canonically, using the terminating expansion whenever two expansions exist. The marker assignments are a declared convention, not an assertion that every exposition uses the same letters. An arbitrary real target would require its entire infinite decimal expansion. A text field cannot supply that, so the live constructor promises only what it can finish exactly.

Why the interval check is finite. Once a closed base cylinder sits inside the open interval, every point obtained by appending digits sits inside too. The page verifies both cylinder inequalities and both point inequalities by BigInt cross-products.

Why this is stronger than Darboux. Mapping every nonempty open interval onto the whole real line implies the intermediate value property. The reverse implication is false: an ordinary Darboux function need not be onto on every interval.

Layer two: the flagship depth layer

Almost nowhere, but inside every room

The digit trick looks indiscriminate only from far away. Fix a valid code stem ending at C. Every later base digit must be an ordinary decimal digit. At each step, ten branches survive and three control branches die.

Operate the exceptional set computing

Tail depth n:

The counts are exact BigInts. Only the decimal rendering of the fraction and the logarithmic dimension estimate use browser floating point.

Admissible cylinders

All cylinders

Exact surviving fraction

Decimal fraction, rounded

Dimension estimate

Dimension equation

The curve plots the computed surviving fraction from the first level through the selected level on a logarithmic vertical scale. It is a finite display of exact counts, not a sample of function values.

For a fixed stem, the limiting continuation is a self-similar set with ten copies, each scaled by one thirteenth. The open set condition gives dimension log(ten) divided by log(thirteen). The nonzero set is a countable union of affine copies of such sets, with lower-dimensional exceptions removed, so it has that same dimension and one-dimensional Lebesgue measure zero.

The same target, repeatedly computing

Distinct witnesses in the chosen interval:

Each witness chooses a different deeper base-13 cylinder, then appends the same target payload. The list reuses the interval and target above.

Dense level sets. Run the constructor in any open interval with the same target. It returns a point there, so every level set is dense. Choosing different deeper cylinders makes as many distinct witnesses as requested.

Dense graph, zero almost everywhere. Every open rectangle contains a constructed graph point, yet the function is nonzero only on the null set just counted. Deliberate witnesses reveal the graph. Uniform random sampling almost surely reveals only the zero line.

Layer three: the open edge

The theorem is closed. The provenance is not.

The existence question here is settled by the explicit constructor: every nonempty open interval maps onto the real line, and continuity at any point is impossible because a sufficiently small neighborhood of a continuous point would have bounded image.

The genuine uncertainty is archival. As of 2026-08-01, the sources reviewed for this page did not identify an original publication by Conway or a reliable invention date. Greg Oman's 2014 paper calls the construction Conway's idea and thanks him. That supports the attribution used here, but it does not establish a priority date.

This is not a century-old converse problem solved by Conway. Darboux's 1875 memoir already states that discontinuous functions can pass through every intermediate value and proves the intermediate value property for finite derivatives. Conway's attributed construction is remarkable because it gives the much stronger everywhere-surjective property explicitly.

Visible apparatus

The check

The browser recomputes this table from the same equations that drive both instruments. The offline verifier independently derives the results, executes this page's own engine in a Node VM, and mutation-tests the checker. Source dates and pagination are transcriptions, not arithmetic, and are separated below.

running live checks

quantity	recomputed live	check

Every free choice and uncertainty

Radix: thirteen, with ordinary digits zero through nine.
Markers: A is positive, B is negative, C is the output decimal point.
Canonical input expansion: choose the terminating form, equivalently forbid a tail eventually all twelves.
Real-line extension: scan the canonical base-thirteen digits of the absolute input. Sign and integer digits are only a finite prefix.
Cylinder choice: the first depth with a complete cylinder, then the leftmost one at that depth.
Payload choice: canonical terminating decimal digits followed by zeros forever.
Rational display: the natural power-of-thirteen denominator is kept unreduced so the digit positions remain visible.
Default demonstration: the prefilled interval and target, depth thirty, and seven witnesses are examples only.
Input scope: at most two hundred fifty-six characters, terminating decimal syntax only.
Fraction display: fifteen places after the decimal point. Exact numerator and denominator remain visible.
Dimension display: twelve places after the decimal, computed with Math.log in binary floating point.
Depth range: one through fifty, chosen for legibility rather than a mathematical limit.
Witness control: two through twenty-four, chosen to keep the exact list readable.
The canvas draws one computed mark per depth. It never samples the function.
No arbitrary-real evaluator is claimed. Finite inspection cannot generally prove that no valid code begins later.
The fractal dimension is inferred by applying Hutchinson's theorem to this digit set. Hutchinson's paper does not discuss Conway's function.
The construction is attributed to Conway. An original Conway publication and reliable invention date were not located.

Transcribed source record

These facts cannot be recomputed from the equations. They were checked against the opened sources listed in the research note.

source	record transcribed
Darboux, Mémoire sur les fonctions discontinues	Annales ENS, series 2, volume 4, 57-112, 1875, DOI 10.24033/asens.122
Oman, The Converse of the Intermediate Value Theorem	Missouri Journal of Mathematical Sciences 26(2), 134-150, 2014, DOI 10.35834/mjms/1418931955
Boylan, Arithmetic Digit Manipulation and The Conway Base-13 Function	arXiv preprint, 2021, arXiv:2103.10249
Hutchinson, Fractals and Self-Similarity	Indiana University Mathematics Journal 30(5), 713-747, 1981, DOI 10.1512/iumj.1981.30.30055