Hello all, Recently I came across the following statement: What happens when a convex function f achieves the value −1 at some point xo? Usually, a degenerate behaviour occurs. For instance, suppose that f is defined on R, and f(0) = − infinity. If f(1) is finite (say), then one must have f(x) = −1 for all 0 <= x < 1 and f(x) = +infinity for all x > 1. Apparently there is no restriction on the function characteristics, e.g. continuity, on f, why is it a "MUST"? If f is continuous (is this allowed for extended real-valued function?), it seems this is not a "MUST". Or please kindly advise me on the definition of extended real-valued function as well as its characteristics. Thanks very much.