A singular function with a non-zero finite derivative on a dense set (Q2637055)

Let \(C^*\) denote the set of real numbers in \([0,1]\) with ternary expansions having the tails that can be written as sequences of pairs of digits, each of which is either \(20\) or \(02\). The authors construct a strictly increasing continuous singular function \(f:[0,1]\to[0,1]\) such that the derivative \(f'(x)\) is finite and non-zero for all \(x\in C^*\). Note that \(C^*\) is a dense uncountable subset of \([0,1]\).