An example of a strictly increasing quasisymmetric function that also a purely singular function can be defined as: \begin{equation*} f(x) = \lim_{k \rightarrow \infty} \int_0^x \prod_{i=1}^k (1 + \lambda \cos n_i s) ds , \end{equation*}where $0< \lambda < 1$ and carefully picked $n_i$ . We can pick the $n_i$ such that $n_{i+1}$ is strictly greater then $\sum_{j=1}^i n_j$ . However if we pick the $\lambda$ and $n_i$ more carefully, we can construct functions with the quasisymmetricity constant as close to 1 as we want. That is, we can construct functions such that \begin{equation*} \frac{1}{M} \leq \frac{f(x+t)-f(x)}{f(x)-f(x-t)} \leq M \end{equation*}for all $x$ and $t$ where $M$ is as close to 1 as we want. If $M=1$ note that the function must be a straight line.

It is also possible from this to construct a quasiconformal mapping of the upper half plane to itself by extending this function to the whole real line and then using the Beurling-Ahlfors quasiconformal extension. Then we'd have a quasiconformal mapping such that its boundary correspondence would be a purely singular function.

For more detailed explanation, graphs and proof (it is too long to reproduce here) see bibliography.

Bibliography

• A. Beurling, L. V. Ahlfors. The boundary correspondence under quasiconformal mappings. Acta Math., 96:125-142, 1956.
• J. Lebl. Quasiconformal Extensions of Quasisymmetric Mappings. Masters thesis, San Diego State University, San Diego, CA, May 2003. Also available at http://www.jirka.org/thesis.pdf