Singular Strictly Monotone Functions

We describe a universal approach to constructing continuous strictly monotone increasing singular functions on the closed interval $[-1,1]$. The “generator” of the method is the series $\sum_{k=1}^\infty\pm2^{-k}$ with random permutation of signs, and the corresponding functions are generated as distribution functions of such series. As examples, we consider two stochastic methods of arranging signs: independent and Markov.