An Epi-Convergence Result for Bivariate Convex Functions

We prove that for a large class of convex and lower semi-continuous biavariate functions defined over $\mathbb{R}^N$, epi-convergence in one variable implies epi-convergence in both variables. We also show that for closed-valued and graph-convex mappings with domains with non empty interiors, pointwise convergence implies graph convergence. We provide a number of applications for both results