The authors introduce a new continuous strictly increasing singular function \(F_{3,2}\) in the following way. They look at \(x\in(0,1]\) with a ternary representation \[ x=0_{\cdot3}0\dots0\alpha_10\dots0\alpha_2\dots0\dots0\alpha_n\dots \] \(\alpha_i\in\{1,2\}\) as in a concatenation of blocks \(B_i^{(3)}\), where \(B_1^{(3)}=2\), \(B_2^{(3)}=1\), \(B_3^{(3)}=02\), \(B_4^{(3)}=01\), \(B_5^{(3)}=002\), \(B_6^{(3)}=001\)…, i.e., \[ x=0_{\cdot3}B_{i_1}^{(3)}B_{i_2}^{(3)}\dots B_{i_n}^{(3)}\dots \tag{1} \] and similarly, using the binary representation, they construct the value of the function as a concatenation of blocks
\(B_k^{(2)}=\overbrace{0\dots01}^{k}\), i.e.,
\[ F_{3,2}(x):=0_{\cdot2}B_{i_1}^{(2)}B_{i_{2}}^{(2)}\dots B_{i_{n}}^{(2)}\dots \]
if \(x\) has the form (1).
The singularity of \(F_{3,2}\) is proved by using the Banach criterion of absolute continuity and Borel’s normal numbers. The proof is clear and interesting. In the introduction the authors give a short history of singular functions and illuminate the main idea of the construction of such functions: We can write a real number using one system of number representation and read the result in some other system.