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\title{Cantor Function}
\author{Todd Mullen}
\date{\today}
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\pagestyle{myheadings} \markboth{Real Analysis}{Todd Mullen}

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\begin{document}
\begin{center}
{\Large\textbf{The Cantor Function $F_c(x)$}}
\end{center}

\noindent A strange construction, and a particular example of a
Devil's staircase, the Cantor Function is defined on the Cantor Set
over the interval $[0,1]$.  Weissenstein's Mathworld gives the
definition, for values in the Cantor Set:
\begin{displaymath}
x=\sum_i 2 \cdot 3^{-n_i},
\end{displaymath} Then the Cantor Function is defined as:
\begin{displaymath}
F_c(x)=\sum_i 2^{-n_i}
\end{displaymath}
It is the only function satisfying the conditions:
\begin{eqnarray}
F(0)=0\\ F\left(\frac{x}{3}\right)=\frac{F(x)}{2}\\ F(1-x)=1-F(x)
\end{eqnarray}
The Cantor Function is real valued and monotone increasing.  This
function is continuous over its domain, but not absolutely
continuous.\footnote{\renewcommand{\baselinestretch}{1}\small\scriptsize
For a set of intervals: $(y_k,x_k)$ with $k = \{1, ..., n\}$
satisfies $\sum_{k=1}^n (y_k-x_k)<\delta$ then $\sum_{k=1}^n
|f(y_k)-f(x_k)|<\varepsilon$
\renewcommand{\baselinestretch}{1.2}normal}
It has no derivatives $\forall c_o\in C$, and $\frac{d}{dx}F(x)=0,
\forall x\in C^c $(the rest of the interval $[0,1]$), but manages
to increase from $F_c(0)=0$ to $F_c(1)=1$.  The following
algorithm calculates function values rather easily:  1) Convert
$x\in C$ to base 3 (using no 1's \emph{if possible}),  2) Replace
the first `$1$' (if there is one) with a `$2$,' everything after
this $2$ becomes $0$,  3) Replace all `2s' with `1s,'  4) Read
this number as binary and viola you have $F_c(x)$  For instance:
$(\frac{3}{4})_{10}\rightarrow(.20202...)_{3}\rightarrow(.10101...)_{2}\rightarrow(\frac{2}{3})_{10}$

\begin{figure*}[h]
	\begin{center}
		\includegraphics[bb= 10 10 150 100]{CantorFunction.eps}
	\end{center}
	\label{fig:CantorFunction}
\end{figure*}


\begin{figure}
	\begin{center}
		\includegraphics[bb=10 10 150 100]{H:00ppffd.eps}
	\end{center}
	\label{fig:00ppffd}
\end{figure}

\begin{figure}
	\begin{center}
		\includegraphics[bb=10 10 400 400]{CantorFunction.eps}
	\end{center}
	\label{fig:CantorFunction}
\end{figure}

\end{document}