The article deals with a class of duality functions given by the solution of a system of functional equations related to the generalized De Rham system \[ f(kx) = k' + (1 - k')f(x), \qquad f(k + (1 - k)x) = k'f(x); \] here is \(k\) and \(k'\) are constant, and \(x \in [0,1]\). It is proved that the only bounded solution \(N_{k,k'}\) of this system is strictly decreasing and continuous; moreover, if \(k + k' \neq 1\) then there exists a set of measure \(1\) in which the derivative of \(N_{k,k'}\) vanishes (and \(N_{k,k'}\) does not admit a non-zero derivative at any \(x \in [0,1]\)). The functions \(N_{k,k'}\) and \(N_{k',k}\) are inverse to each other. At last, \(N_{k,k'}\) applies a set of Lebesgue measure \(1\) onto a set of Lebesgue measure \(0\) whose Hausdorff dimension is \(\ln [k^k(1 - k)^{1-k}] / [\ln [{k'}^{1-k}(1 - k')^k]\). In the end of the article, similar results are obtained for the function \(N_k\) that is the unique bounded solution of the system \[ f(kx) = k + (1 - k)f(x), \qquad f(k + (1 - k)x) = kf(x) \] (really, this system is a partial case of the previous system when \(k + k' = 1\)). The basic role in the authors’ argument plays the dyadic representation of numbers from \((0,1]\) of the type \(x = \sum_{d=0}^\infty (1 - k)^d k^{m_d}\) (\(1 \leq m_0 \leq m_1 \leq \dots \leq m_d \leq \dots\)). In particular, this dyadic representation allows the authors to obtain an explicit expressions for the functions \(N_{k,k'}\) and \(N_k\).