Numeral systems with non-zero redundancy and their applications in the theory of locally complex functions

In this paper we study representations of real numbers in a numeral system with the base $a>1$ and alphabet (digits set) $A\equiv\{0,1,...,r\}$, $a-1<r\in N$ given by $$x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n}\equiv \Delta^{r_a}_{\alpha_1\alpha_2...\alpha_n...}, \alpha_n\in A.$$ Since the alphabet is redundant the numbers from the interval $[0;\frac{r}{a-1}]$ have not a single representation and can even have a continuous set of different representations. We describe the geometry (topological and metric properties) of such representations (the $r_a$-representations) in terms of cylinders defined by $$\Delta^{r_a}_{c_1c_2...c_m}= \{x: x=\Delta^{r_a}_{c_1c_2...c_ma_1a_2...a_n...}, a_n\in A\},$$ We analyze their properties in detail, including the specific nature of overlaps. We present results on the structural, variational, topological, metric and partially fractal properties of the function defined by $$f\left(x=\sum_{n=1}^{\infty}\frac{\alpha_n}{(r+1)^n}\right)= \Delta^{r_a}_{\alpha_1\alpha_2...\alpha_n...},\alpha_n \in A.$$ We prove the function is continuous at all points of the interval $[0,1]$ that have a unique representation in the classical numeral system on the base $r+1$ and prove the function is discontinuous at points of a countable everywhere dense set in $[0,1]$. Furthermore, we show that the function is nowhere monotonic and has unlimited variation. In the particular case $r=1$ and $a=\frac{1+\sqrt{5}}{2}$, we specify fractal level sets with Hausdorff--Besicovitch dimension not less than $-\log_a2$.