Proposition 21.11.18B6B15657CD64

Proof. For proper closed functions this is the uniqueness assertion in the convex closure theorem. More explicitly, the closure of the epigraph over the relative interior of the effective domain is the whole epigraph: for a point of the epigraph, take convex combinations with a fixed epigraph point whose base is in that relative interior, and let the coefficient of the fixed point tend to zero. Thus identical values on the common relative interior give identical epigraphs and hence identical functions.

The improper cases in the definition of closedness are also determined: empty effective domain means the function identically $+\infty$; if one function is identically $-\infty$, the common relative interior is $N$ and the value agreement makes the other identical to it. These cases complete the proof.