ADDITION THEOREMS AND BINARY EXPANSIONS

Let an interval I subset of R and subsets D-0, D-1 subset of I with D- 0 boolean OR D-1 = I and D-0 boolean AND D-1 = 0 be given, as well as functions r(0):D-0 --> I, r(1):D-1 --> I. We investigate the system (S ) of two functional equations for an unknown function f: I --> [0, 1]: 2f(x) = f(r(0)(x)) if x is an element of D-0, 2f(x) - 1 = f(r(1)(x)) if x is an element of D-1. We derive conditions for the existence, con tinuity and monotonicity of a solution. It turns out that the binary e xpansion of a solution can be computed in a simple recursive way. This recursion is algebraic for, e.g., inverse trigonometric functions, bu t also for the elliptic integral of the first kind. Moreover, we use ( S) to construct two kinds of peculiar functions: surjective functions whose intervals of constancy are residual in I, and strictly increasin g functions whose derivative is 0 almost everywhere.