In this dissertation, a new method for obtaining convexity and optimization results for functions with integer and real (i.e. mixed) variables is introduced and this new method is applied to obtain mixed convexity and optimization results for some of the known mixed variable functions in the literature. These mixed variable functions include the Erlang delay and loss formulae in telecommunication systems, an (S-1,S) inventory model (suggested by Das (1977)), and an M/Ek/1 queueing system model (suggested by Kumin (1973)). Local and global mixed convexity and optimization results for these mixed variable functions are obtained after introducing definitions for a condense discrete convex set, a condense discrete convex function, a discrete Hessian matrix, a mixed convex set, a mixed convex function, and a mixed Hessian matrix. Symbolic toolbox of MATLAB R2009a is used to obtain symbolic results. Computational discrete and mixed convexity and optimization results are also obtained by using MATLAB R2009a. The results obtained in this work are important because prior to this work no joint convexity results for mixed functions for mixed functions have been defined. This dissertation obtains such joint results. In addition, for real variable functions that are strictly convex, it is well-known that any local minimum is also the global minimum. In this work, similar results are obtained for mixed strictly convex functions. A new Hessian matrix defined for mixed variable functions can be used to determine whether any local minimum is also the global minimum.