The Cantor Set and the Cantor-Lebesgue Function

Let $f:[a, b] \rightarrow \mathbb {R}$ be continuous and strictly increasing. Show that $f^{-1}:[f(a), f(b)] \rightarrow \mathbb {R}$ is continuous.

Let $f:[a, b] \rightarrow \mathbb {R}$ be continuous. Show that $f$ maps closed sets to closed sets and maps $F_{\sigma }$ sets to $F_{\sigma }$ sets. Does $f$ map measurable sets to measurable sets?

Let the function $f:[a, b] \rightarrow \mathbb {R}$ be Lipschitz, that is, there is a constant $c \geq 0$ such that for all $u, v \in [a, b],\left|f(u)-f(v)\right| \leq c\left|u-v\right|$. Show that $f$ is continuous and maps sets of measure zero to sets of measure zero. Conclude that $f$ maps measurable set to measurable sets.

Let $F$ be the subset of $[0,1]$ constructed in the same manner as the Cantor set except that each of the intervals removed at the $n$th deletion stage has length $\alpha 3^{-n}$ with $0 < \alpha < 1$. Show that $F$ is a closed set, $[0,1] \sim F$ is dense in $[0,1]$, and $m(F) = 1-\alpha$. Such a set $F$ is called a generalized Cantor set.

Show that there is an open set of real numbers which, contrary to intuition, has a boundary of positive measure.

A subset $A$ of $\mathbb {R}$ is said to be nowhere dense in $\mathbb {R}$ provided that the closure of every open set $\mathcal{O}$ has a non-empty open subset that is disjoint from $A$. Show that the Cantor set is nowhere dense in $[0,1]$.

Let $f:[a, b] \rightarrow \mathbb {R}$ be continuous and $B$ be a Borel set. Show that $f^{-1}(B)$ is a Borel set.