Domain of a convex function is convex

Corollary. If $f:X\to\mathbb{R}$ is a convex function on a vector space $X$, then its domain $\mathrm{dom}(f)$ is a convex set .

Connection. This follows immediately from Jensen’s inequality in Theorem (equivalent characterizations) : if $x,y\in\mathrm{dom}(f)$, then the right-hand side is finite, forcing $f(\lambda x+(1-\lambda)y)<\infty$.