Understanding defn.of the Cantor- Lebesgue fn. $\varphi$ on the Cantor set $\textbf{C}.$

Question

Here is the definition of $\varphi$ from Royden "Real Analysis" fourth edition:

But I do not understand the definition of the Cantor- Lebesgue function $\varphi$ on the cantor set, can someone explain it to me with a numeric example please?

Answer 1

Any $x\in [0,1]$ is in $C$ iff $x$ has a (unique) representation in base $3$ that does $not$ use the digit $1.$ E.g. in ternary, $1/3=0.0\overline 2 =0.02222222....$ For any $x\in C$ we change all the $2$'s in this representation to $1$'s and interpret the result as a digit-sequence in base $2$ (not base $3$) and this gives us $\varphi (x).$

The end-points $a,b$ of a "removed" open interval $(a,b)$ are points of $C,$ and $\varphi (a)=\varphi (b).$ And $(a,b)$ is disjoint from $C,$ so we can let $\varphi (x')=\varphi (a)=\varphi (b)$ for all $x'\in (a,b).$

Example: $(a,b)=(19/27, 20/27).$ In base $3$ we have $a=0.200\overline 2$ and $b=0.202\overline 0.$ Changing the $2$'s to $1$'s produces the sequences $0.100\overline 1$ and $0.101\overline 0.$ In base $2$ these are $both$ representations of $1/2+1/8=5/8$, so $\varphi (a)=\varphi (b)=5/8.$ And $\varphi (x')=5/8$ for all $x'\in (a,b).$