I have encountered this function while trying to construct a function that is discontinuous everywhere but has IVP.But the function is not clear to me.I need a proper explanation about the definition and graph of the function
2026-03-26 06:28:35.1774506515
Can someone give me information about conway base 13 function and give me a graph of the function?
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The Conway base $13$ function is a artificial function that encodes decimal representations as base $13$ representations. If the base $13$ representation of $x$ encodes the decimal representation of $y$ then $f(x)=y$. If the base $13$ representation of $x$ does not encode any decimal representation (which is the case for "most" numbers) then $f(x)=0$.
The encoding is constructed in such a way that between any pair of real numbers there is an encoding (in fact, an infinite number of encodings) of every possible decimal representation. Since every real number $y$ has a decimal representation, this means that in any interval $[a,b]$ there is some $c \in [a,b]$ such that $f(c)=y$. And this implies the IVP.
The only significance of $13$ is that it is $3$ greater than $10$, thus providing $3$ additional digits $A,B,C$ (or $+, -, .$ in Conway's original formulation) on top of the decimal digits $0 \dots 9$ to support the encoding rules.