According to the definition of a convex function as stated in the book `Convex Optimization’ by Boyd and Vandenberghe (p. 67) a function $f\colon \mathbb{R}^{n} \rightarrow \mathbb{R}$ is convex if $\rm{dom}(f)$ is convex and if
(*): $f(\alpha \textbf{x}+(1-\alpha)\textbf{y})\leq \alpha f( \textbf{x})+(1-\alpha)f(\textbf{y})$ for all $\alpha\in[0,1]$ and $\textbf{x}$, $\textbf{y} \in \rm{dom}(f)$.
So, my actual question: when attempting to show a function $f$ is convex, should I also show that its domain is convex, or is showing that the property (*) is satisfied sufficient?