TITLE: Example of continuous but not absolutely continuous strictly increasing function
QUESTION [8 upvotes]: Could one give an example of a strictly increasing, continuous but not absolutely continuous, function on $[0,1]$ into $[0,1]$ or on $[0,1)$ into $R$ or any of the related combinations of 1-d domain and range?

REPLY [4 votes]: Let $f:[0,1)\to \mathbb{R}$, $f(x)=\tan(\pi x/2)$. This function is continuous and strictly increasing but not absolutely continuos.
Just to show that this function is in fact not absolutely continuous. Take $\epsilon=1$, and suppose there is a $\delta>0$ such that whenever a finite sequence of pairwise disjoint sub-intervals $(x_{k},y_{k})$ of $[0,1)$ satisfies $\sum_{k}|x_k - y_k| < \delta$ then we have $\sum_{k} |f(y_k)-f(x_k)| < 1$.
Since $\lim_{x \to 1-}f(x) = +\infty$ and $f$ is continuous, let $x_0 \in [1-\delta,1)$ so we can get $y_0 \in [1-\delta,1)$ such that $f(y_0)-f(x_0)>1$.
Using only the interval $(x_0,y_0)$ to test the definition of absolute continuity we have then that $|y_0 - x_0|<\delta$ but $|f(y_0) - f(x_0)|>1$. Therefore, $f$ is not absolutely continuous.