  Integrally convex functions constitute a fundamental function class in
discrete convex analysis. This paper shows that an integer-valued integrally
convex function admits an integral subgradient and that the integral
biconjugate of an integer-valued integrally convex function coincides with
itself. The proof is based on the Fourier-Motzkin elimination. The latter
result provides a unified proof of integral biconjugacy for various classes of
integer-valued discrete convex functions, including L-convex, M-convex,
L$_{2}$-convex, M$_{2}$-convex, BS-convex, and UJ-convex functions as well as
multimodular functions. Our results of integral subdifferentiability and
integral biconjugacy make it possible to extend the theory of discrete DC
(difference of convex) functions developed for L- and M-convex functions to
that for integrally convex functions, including an analogue of the
Toland--Singer duality for integrally convex functions.
