example of a strictly increasing quasisymmetric singular function
An example of a strictly increasing quasisymmetric function that also a purely singular function can be defined as:
$$f(x)=\underset{k\to \mathrm{\infty}}{lim}{\int}_{0}^{x}\prod _{i=1}^{k}(1+\lambda \mathrm{cos}{n}_{i}s)ds,$$ 
where $$ and carefully picked ${n}_{i}$. We can pick the ${n}_{i}$ such that ${n}_{i+1}$ is strictly greater then ${\sum}_{j=1}^{i}{n}_{j}$. However if we pick the $\lambda $ and ${n}_{i}$ more carefully, we can construct functions with the quasisymmetricity constant as close to 1 as we want. That is, we can construct functions such that
$$\frac{1}{M}\le \frac{f(x+t)f(x)}{f(x)f(xt)}\le M$$ 
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 BeurlingAhlfors quasiconformal extension. Then we’d have a quasiconformal mapping such that its boundary correspondence would be a purely singular function.
For more detailed explanation, and proof (it is too long to reproduce here) see bibliography.
Bibliography

•
A. Beurling, L. V. Ahlfors. . Acta Math., 96:125142, 1956.

•
J. Lebl. . . Also available at http://www.jirka.org/thesis.pdfhttp://www.jirka.org/thesis.pdf
Title  example of a strictly increasing quasisymmetric singular function 

Canonical name  ExampleOfAStrictlyIncreasingQuasisymmetricSingularFunction 
Date of creation  20130322 14:10:37 
Last modified on  20130322 14:10:37 
Owner  jirka (4157) 
Last modified by  jirka (4157) 
Numerical id  5 
Author  jirka (4157) 
Entry type  Example 
Classification  msc 26A30 