The Minkowski function \(?(x): [0,1]\to [0,1]\) is strictly increasing, continuous, and maps the rational numbers onto the dyadic rationals. \textit{R. Salem} has proved in 1943 [Trans. Am. Math. Soc. 53, 427--439 (1943; Zbl 0060.13709)] that if \(x\in[0,1]\) has a continued fraction expansion with unbounded partial quotients and if \(?'(x)\) exists and is finite, then \(?'(x)=0\). This shows that \(?(x)\) is a singular function. In the present paper, the authors improve this long-standing result significantly by showing that existence of \(?'(x)\) in \(\mathbb R\) implies \(?'(x)=0\) for any \(x\in[0,1]\). They also give explicit conditions (some in terms of the continued fraction expansion of \(x\) and some in terms of the alternated dyadic expansion of \(?(x)\)) to determine whether \(?'(x)\) is 0 or infinite (if it exists in a wide sense).