This paper deals with convexity for discrete functions. A piecewise-convex extension for a class of discrete functions (integer-valued functions defined on integer lattice points) is defined and a function is said to be integrally convex if its piecewise-convex extension is globally convex. In the introduction of the paper the known relation to submodular functions and to L-convex functions is summarized. Also a reference is given that M- and L-convex functions are in a one-to-one correspondence. In the second section of the paper new results clarifying the relationship between L-convex functions and submodular integrally convex functions are given. The third section contains implications of these results concerning the concepts of conjugacy and duality (including separation results). The last section of the paper gives alternative proofs for the separation theorems relating them to the ordinary separation theorems from convex analysis. The paper is well written and structured.