Why should we define improper convex function so?

Question

Now I'm reading Rockafellar's book. I don't understand why he defines improper convex function as $+\infty$ outside $\mbox{cl}(\mbox{dom} f)$. From page 54:

Why isn't it $-\infty$?

Answer 1

It's the only value you can use to preserve convexity everywhere outside the domain of $f$.

By the definition of convexity we know that $$f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)y $$ for $x,y \in \mbox{dom}(f)$ and $\lambda \in [0,1]$. When we want to extend $f$ outside of its domain (this looks like we're assigning values to $f$ where it didn't have any previously, but actually we're finding a new function $f^*$ that coincides with $f$ on $\mbox{dom}(f)$ and has values elsewhere additionally) we still need to preserve the convexity relation. That means $$f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)y $$ still needs to hold when $x\in \mbox{dom}(f)$ and $y\not\in \mbox{dom}(f)$ -- and the only way you can guarantee that is to make $f(y)$ bigger than any other number.