Continuous Functions with Locally Complicated and Fractal Properties Related to Infinite-Symbol B-Representation of Numbers

We introduce and study a massive class of continuous functions defined in the interval (0; 1) by using a special encoding (representation) of the argument with an alphabet Z = {0, ±1, ±2, . . .}: \(\begin{array}{c}x={b}_{{\alpha }_{1}}+\sum\limits_{k=2}^{m}{b}_{{\alpha }_{k}}\prod\limits_{i=1}^{k-1}{\Theta }_{{\alpha }_{i}}\equiv {\Delta }_{{\alpha }_{1}{\alpha }_{2}\dots {\alpha }_{m}\left(\mathrm{\varnothing }\right)}^{B},\\ x={b}_{{\alpha }_{1}}+\sum\limits_{k=2}^{\infty }{b}_{{\alpha }_{k}}\prod\limits_{i=1}^{k-1}{\Theta }_{{\alpha }_{i}}\equiv {\Delta }_{{\alpha }_{1}{\alpha }_{2}\dots {\alpha }_{n}\dots }^{B},\end{array}\) $$\begin{array}{c}x=b_{{\alpha }_1}+\sum _{k=2}^m{b}_{{\alpha }_k}\prod _{i=1}^{k-1}{\mathrm{\Theta }}_{{\alpha }_i}\equiv {\mathrm{\Delta }}_{{\alpha }_1{\alpha }_2\cdots {\alpha }_m\left(\varnothing \right)}^B,\\ x=b_{{\alpha }_1}+\sum _{k=2}^{\infty }{b}_{{\alpha }_k}\prod _{i=1}^{k-1}{\mathrm{\Theta }}_{{\alpha }_i}\equiv {\mathrm{\Delta }}_{{\alpha }_1{\alpha }_2\cdots {\alpha }_n\cdots }^B,\end{array}$$

where α_n ∈ Z‚ Θ_n > 0 ∀n ∈ Z; \({\sum }_{n=-\infty }^{+\infty }{\Theta }_{n}=1\) ${\sum }_{n=-\infty }^{+\infty }{\mathrm{\Theta }}_n=1$ , and \({b}_{n+1}\equiv {\sum }_{i=-\infty }^{n-1}={b}_{n}+{\Theta }_{n}\forall n\in Z\) $b_{n+1}\equiv {\sum }_{i=-\infty }^{n-1}=b_n+{\mathrm{\Theta }}_n\forall n\in Z$ .

A function f; which is the main object of our investigations, is defined by the following equalities: \(\left\{\begin{array}{l}f\left(x={\Delta }_{{i}_{1}\dots {i}_{k}\dots }^{B}\right)={\sigma }_{{i}_{1}1}+{\sum }_{k=2}^{\infty }{\sigma }_{{i}_{k}k}{\prod }_{j=1}^{k-1}{p}_{{i}_{j}j}\equiv {\Delta }_{{i}_{1}\dots {i}_{k}\dots },\\ f\left(x={\Delta }_{{i}_{1}\dots {i}_{m}\left(\mathrm{\varnothing }\right)}^{B}\right)={\sigma }_{{i}_{1}1}+{\sum }_{k=2}^{m}{\sigma }_{{i}_{k}k}{\prod }_{j=1}^{k-1}{p}_{{i}_{j}j}\equiv {\Delta }_{{i}_{1}\dots {i}_{m}\left(\mathrm{\varnothing }\right)},\end{array}\right.\) $$\left\{\begin{array}{c}f\left(x,=,{\mathrm{\Delta }}_{i_1\cdots i_k\cdots }^B\right)={\sigma }_{i_{1}1}+{\sum }_{k=2}^{\infty }{\sigma }_{i_{k}k}{\prod }_{j=1}^{k-1}{p}_{i_{j}j}\equiv {\mathrm{\Delta }}_{i_1\cdots i_k\cdots },\hfill \\ f\left(x,=,{\mathrm{\Delta }}_{i_1\cdots i_m\left(\varnothing \right)}^B\right)={\sigma }_{i_{1}1}+{\sum }_{k=2}^m{\sigma }_{i_{k}k}{\prod }_{j=1}^{k-1}{p}_{i_{j}j}\equiv {\mathrm{\Delta }}_{i_1\cdots i_m\left(\varnothing \right)},\hfill \end{array}\right)$$

where the infinite matrix ||p_ik||, i ∈ Z; k ∈ N, satisfies the conditions \(\begin{array}{l}\left.1\right) \left|{p}_{ik}\right|<1 \forall i\in Z, \forall k\in N;\\ \left.2\right) {\sum }_{i\in Z}{p}_{ik}=1\forall k\in N;\\ \left.3\right) 0<{\sum }_{k=2}^{\infty }{\prod }_{j=1}^{k-1}{p}_{{i}_{j}j}<\infty \forall \left({i}_{j}\right)\in L;\\ \left.4\right) 0<{\sigma }_{ik}\equiv {\sum }_{j=-\infty }^{i-1}{p}_{jk}<1\forall i\in Z,\forall k\in N.\end{array}\) $$\begin{array}{c}\left(1\right)\left|p_{\mathit{ik}}\right|<1\forall i\in Z,\forall k\in N;\hfill \\ \left(2\right){\sum }_{i\in Z}{p}_{\mathit{ik}}=1\forall k\in N;\hfill \\ \left(3\right)0<{\sum }_{k=2}^{\infty }{\prod }_{j=1}^{k-1}{p}_{i_{j}j}<\infty \forall \left(i_j\right)\in L;\hfill \\ \left(4\right)0<{\sigma }_{\mathit{ik}}\equiv {\sum }_{j=-\infty }^{i-1}{p}_{\mathit{jk}}<1\forall i\in Z,\forall k\in N.\hfill \end{array}$$

The introduced class of functions contains monotone, nonmonotone, and nowhere monotone functions, functions without monotonicity intervals except the constancy intervals, Cantor-type and quasi-Cantor-type functions, and functions of bounded and unbounded variation. We establish criteria for the function f to be a monotonic and Cantor-type function, as well as a criterion of nowhere monotonicity of this function. We obtain expressions for the Lebesgue measure of the set of nonconstancy of this function and for the variation of the function. The necessary and sufficient conditions under which the function has unbounded variation are established. For a particular case, we describe the self-similarity (structural fractality) of the plot of the analyzed function and study its differential properties.