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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

  • The graph of the function looks like this: https://i.sstatic.net/0zmOf.png –  Jun 05 '19 at 11:58
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    Have you read the Wikipeaid article on the Conway base 13 function? If not, go and read it now. And then please explain what it is that you don't understand. We are not mind-readers! – TonyK Jun 05 '19 at 12:08
  • I've voted to close because this is a duplicate of one of the very earliest questions on the site. There are other questions about it too, found by searching "Conway base 13" – Mark Bennet Jun 05 '19 at 12:15
  • Rahul I opened the above link but there is no graph only a black screen.Please can you once again send a graph of conway base 13. –  Jun 06 '19 at 12:07
  • Kishalay, that is really the graph of the function! It takes all values on every interval, so its graph is dense everywhere. @Rahul was making a little joke, I think, but one with a point to it. – TonyK Jun 07 '19 at 10:11

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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.

gandalf61
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