Theorem regarding convex sublevel sets defined by convex functions

Question

It is known and it is easy to prove that given a convex function $f : \mathbb{R} \to \mathbb{R}$, then the sublevel sets $\left\lbrace x \middle| f(x) < a\right\rbrace$ and $\left\lbrace x \middle| f(x) \leq a\right\rbrace$ with $a \in \mathbb{R}$ are convex sets, but it can not find the actual theorem so i can reference it.

Please do forgive me for asking such question but i am in terrible need of it.

Thanks in advance.

Answer 1

Section 3.1.6 in 'Convex Optimization' by Boyd and Vandenberghe provides a proof (although it is not presented as an explicit theorem).