An increasing embedding f of an interval of the real line is quasisymmetric if there is \(q\geq 1\) such that \(1/q\leq (f(x+t)- f(x))/(f(x)-f(x-t))\leq q\) for all distinct x, \(x+t\), x-t. The paper gives an example of a quasisymmetric map f of the unit interval I with the property that there is a measurable subset Y of I such that the Hausdorff dimensions of both \(I\setminus Y\) and f(Y) are less than a given positive constant. There is also a quasisymmetric map of the real line with these properties. The map f is a modification of the singular function of \textit{R. Salem} in ``On some singular monotonic functions which are strictly increasing'' [Trans. Am. Math. Soc. 53, 427-439 (1943; Zbl 0060.137)].