**Abstract:** The present work summarizes and further develops properties of orbit functions. Orbit functions, based on sign homomorphism of Weyl groups, are defined. For each Weyl group of two different root lenghts there are four sign homomorphisms, which lead to ten types of orbit functions ( E-,S-,C- functions). Proof of their continuous orthogonality and completness relation on the fundamental domain is given. In the key part of the work, we discretize the fundamental domains and prove the discrete orthogonality relation and completness relation of orbit functions on the discretized fundamental domain.