Convex Function on a Convex Subset of $\mathbb{R}^n$

In the setting of Convex Functions on $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$ be convex and let $f:C\to\mathbb{R}$ be a function.

(i) Convex function. $f$ is convex if

$$f\bigl((1-t)x+ty\bigr)\le(1-t)f(x)+tf(y)\qquad\text{for all }x,y\in C\text{ and }t\in[0,1],$$

the point $(1-t)x+ty$ lying in $C$ since it is in the segment $[x,y]$, contained in $C$ by Segments and Convex Subsets of $\mathbb{R}^n$ (ii).