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

Let $n\ge1$ be a natural number, let $\mathbb{R}$ be the real numbers, let $C\subseteq\mathbb{R}^n$ be convex, and let $f:C\to\mathbb{R}$.

We say that $f$ is convex on $C$ if for all $x,y\in C$ and every $t\in\mathbb{R}$ with $0\le t$ and $t\le 1$,

$$f\bigl(t\,x+(1-t)\,y\bigr)\le t\,f(x)+(1-t)\,f(y),$$

the point $t\,x+(1-t)\,y$ lying in $C$ because $C$ is convex.