Conway's base 13 function

<h2 id="purpose">Purpose</h2>
Conway's base 13 function was created as part of a "produce" activity: in this case, the challenge was to produce a simple-to-understand function which takes on every real value in every interval, that is, it is an everywhere <a href="/facts/Surjective_function/KezhLpCb">surjective function</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> It is thus <a href="/facts/Nowhere_continuous_function/XbxlDzx1">discontinuous at every point</a>.

<h2 id="sketch-of-definition">Sketch of definition</h2>
<ul><li>Every real number 
 
 
 
 x
 
 
 {\displaystyle x}
 
 can be represented in <a href="/facts/Base_13/RgfoI90F">base 13</a> in a unique canonical way; such representations use the digits 0–9 plus three additional symbols, say {A, B, C}. For example, the number 54349589 has a base-13 representation B34C128.</li>
<li>If instead of {A, B, C}, we judiciously choose the symbols {+, −, .}, some numbers in base 13 will have representations that look like well-formed decimals in base 10: for example, the number 54349589 has a base-13 representation of −34.128. Of course, most numbers will not be intelligible in this way; for example, the number 3629265 has the base-13 representation 9+0−−7.</li>
<li>Conway's base-13 function takes in a real number x and considers its base-13 representation as a sequence of symbols {0, 1, ..., 9, +, −, .}. If from some position onward, the representation looks like a well-formed decimal number r, then f(x) = r. Otherwise, f(x) = 0. (Well-formed means that it starts with a + or − symbol, contains exactly one decimal-point symbol, and otherwise contains only the digits 0–9). For example, if a number x has the representation 8++2.19+0−−7+3.141592653, then f(x) = +3.141592653.</li></ul>
<h2 id="definition">Definition</h2>
Conway defined his base-13 function 
 
 
 
 f
 :
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle f:\mathbb {R} \to \mathbb {R} }
 
 as follows. Write the argument 
 
 
 
 x
 
 
 {\displaystyle x}
 
 value as a tridecimal (a "decimal" in <a href="/facts/Base_13/RgfoI90F">base 13</a>) using 13 symbols as "digits": 0, 1, ..., 9, A, B, C; there should be no trailing C recurring. There may be a leading sign, and somewhere there will be a tridecimal point to separate the integer part from the fractional part; these should both be ignored in the sequel. These "digits" can be thought of as having the values 0 to 12 respectively; Conway originally used the digits "+", "−" and "." instead of A, B, C, and underlined all of the base-13 "digits" to clearly distinguish them from the usual base-10 digits and symbols.

<ul><li>If from some point onwards, the tridecimal expansion of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is of the form 
 
 
 
 A
 
 x
 
 1
 
 
 
 x
 
 2
 
 
 …
 
 x
 
 n
 
 
 C
 
 y
 
 1
 
 
 
 y
 
 2
 
 
 …
 
 
 {\displaystyle Ax_{1}x_{2}\dots x_{n}Cy_{1}y_{2}\dots }
 
 where all the digits 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
 and 
 
 
 
 
 y
 
 j
 
 
 
 
 {\displaystyle y_{j}}
 
 are in 
 
 
 
 {
 0
 ,
 …
 ,
 9
 }
 ,
 
 
 {\displaystyle \{0,\dots ,9\},}
 
 then 
 
 
 
 f
 (
 x
 )
 =
 
 x
 
 1
 
 
 …
 
 x
 
 n
 
 
 .
 
 y
 
 1
 
 
 
 y
 
 2
 
 
 …
 
 
 {\displaystyle f(x)=x_{1}\dots x_{n}.y_{1}y_{2}\dots }
 
 in usual <a href="/facts/Base-10/7omfVNIM">base-10</a> notation.</li>
<li>Similarly, if the tridecimal expansion of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 ends with 
 
 
 
 B
 
 x
 
 1
 
 
 
 x
 
 2
 
 
 …
 
 x
 
 n
 
 
 C
 
 y
 
 1
 
 
 
 y
 
 2
 
 
 …
 ,
 
 
 {\displaystyle Bx_{1}x_{2}\dots x_{n}Cy_{1}y_{2}\dots ,}
 
 then 
 
 
 
 f
 (
 x
 )
 =
 −
 
 x
 
 1
 
 
 …
 
 x
 
 n
 
 
 .
 
 y
 
 1
 
 
 
 y
 
 2
 
 
 …
 .
 
 
 {\displaystyle f(x)=-x_{1}\dots x_{n}.y_{1}y_{2}\dots .}
 
</li>
<li>Otherwise, 
 
 
 
 f
 (
 x
 )
 =
 0.
 
 
 {\displaystyle f(x)=0.}
 
</li></ul>
For example:

<ul><li>
 
 
 
 f
 (
 
 12345
 A
 3
 C
 14.159
 
 …
 
 
 
 13
 
 
 )
 =
 f
 (
 
 A
 3
 C
 14.159
 
 …
 
 
 
 13
 
 
 )
 =
 3.14159
 …
 ,
 
 
 {\displaystyle f(\mathrm {12345A3C14.159} \dots _{13})=f(\mathrm {A3C14.159} \dots _{13})=3.14159\dots ,}
 
</li>
<li>
 
 
 
 f
 (
 
 
 B
 1
 C
 234
 
 
 13
 
 
 )
 =
 −
 1.234
 ,
 
 
 {\displaystyle f(\mathrm {B1C234} _{13})=-1.234,}
 
</li>
<li>
 
 
 
 f
 (
 
 
 1
 C
 234
 A
 567
 
 
 13
 
 
 )
 =
 0.
 
 
 {\displaystyle f(\mathrm {1C234A567} _{13})=0.}
 
</li></ul>
<h2 id="properties">Properties</h2>
<ul><li>According to the intermediate-value theorem, every continuous real function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has the intermediate-value property: on every interval (a, b), the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 passes through every point between 
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 and 
 
 
 
 f
 (
 b
 )
 .
 
 
 {\displaystyle f(b).}
 
 The Conway base-13 function shows that the converse is false: it satisfies the intermediate-value property, but is not continuous.</li>
<li>In fact, the Conway base-13 function satisfies a much stronger intermediate-value property—on every interval (a, b), the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 passes through every real number. As a result, it satisfies a much stronger discontinuity property— it is discontinuous everywhere.</li>
<li>From the above follows even more regarding the discontinuity of the function - its graph is dense in 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
</li>
<li>To prove that Conway's base-13 function satisfies this stronger intermediate property, let (a, b) be an interval, let c be a point in that interval, and let r be any real number. Create a base-13 encoding of r as follows: starting with the base-10 representation of r, replace the decimal point with C and indicate the sign of r by prepending either an A (if r is positive) or a B (if r is negative) to the beginning. By definition of Conway's base-13 function, the resulting string 
 
 
 
 
 
 
 r
 ^
 
 
 
 
 
 {\displaystyle {\hat {r}}}
 
 has the property that 
 
 
 
 f
 (
 
 
 
 r
 ^
 
 
 
 )
 =
 r
 .
 
 
 {\displaystyle f({\hat {r}})=r.}
 
 Moreover, any base-13 string that ends in 
 
 
 
 
 
 
 r
 ^
 
 
 
 
 
 {\displaystyle {\hat {r}}}
 
 will have this property. Thus, if we replace the tail end of c with 
 
 
 
 
 
 
 r
 ^
 
 
 
 ,
 
 
 {\displaystyle {\hat {r}},}
 
 the resulting number will have f(c') = r. By introducing this modification sufficiently far along the tridecimal representation of 
 
 
 
 c
 ,
 
 
 {\displaystyle c,}
 
 you can ensure that the new number 
 
 
 
 
 c
 ′
 
 
 
 {\displaystyle c'}
 
 will still lie in the interval 
 
 
 
 (
 a
 ,
 b
 )
 .
 
 
 {\displaystyle (a,b).}
 
 This proves that for any number r, in every interval we can find a point 
 
 
 
 
 c
 ′
 
 
 
 {\displaystyle c'}
 
 such that 
 
 
 
 f
 (
 
 c
 ′
 
 )
 =
 r
 .
 
 
 {\displaystyle f(c')=r.}
 
</li>
<li>Conway's base-13 function is therefore discontinuous everywhere: a real function that is continuous at x must be locally bounded at x, i.e. it must be bounded on some interval around x. But as shown above, the Conway base-13 function is unbounded on every interval around every point; therefore it is not continuous anywhere.</li>
<li>Conway's base-13 function maps <a href="/facts/Almost_all/KvHxUCCc">almost all</a> the reals in any interval to 0.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Darboux_function/uYXRzISn">Darboux function</a> – All derivatives have the intermediate value propertyPages displaying short descriptions of redirect targets</li></ul>

<ul><li>Oman, Greg (2014). <a href="https://faculty.uccs.edu/goman/wp-content/uploads/sites/15/2021/02/Converse-of-IVT.pdf">"The Converse of the Intermediate Value Theorem: From Conway to Cantor to Cosets and Beyond"</a> (PDF). Missouri J. Math. Sci. 26 (2): 134–150. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.35834%2Fmjms%2F1418931955">10.35834/mjms/1418931955</a>. <a href="https://web.archive.org/web/20160820112316/https://www.uccs.edu/Documents/goman/Converse%20of%20IVT.pdf">Archived</a> (PDF) from the original on 2016-08-20.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bernardi, Claudio (February 2016). "Graphs of real functions with pathological behaviors". Soft Computing. 11: 5–6. arXiv:1602.07555. Bibcode:2016arXiv160207555B. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Stein, Noah. "Is Conway's base-13 function measurable?". mathoverflow. Retrieved 6 August 2023. <a href="https://mathoverflow.net/questions/32641/is-conways-base-13-function-measurable" target="_blank">https://mathoverflow.net/questions/32641/is-conways-base-13-function-measurable</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Conway's base 13 function open-in-new

Conway's base 13 function