I wonder whether maximum of pseudoconvex function is pseudoconvex function or not, as we concern in smooth situation and nonsmooth as well.
As far as I know, maximum of convex function over $\mathbb{R}^n$ is convex function. And this is still true when we take max pointwise towards quasiconvex functions. However, the problem comes when we observe the maximum of some pseudoconvex functions. Can we conclude that the maximum of these things is a pseudoconvex function? Or even when each component is continuously differentiable?
Can someone give me a few ituitive ideas for solving this question and some references to this problem? Thanks a lot!