Conjugacy equations arise from the problem of identifying dynamical systems from the topological point of view. It is well known that when conjugacies exist they cannot, in general, be expected to be smooth. We show that even in the simplest cases, e.g. piecewise affine maps, solutions of functional equations arising from conjugacy problems may have exotic properties: they may be singular, fractal or even everywhere discontinuous. We provide an explicit formula showing how, in certain cases, a solution can be constructively determined. We find, for the same equation, remarkably distinct solutions.