Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval

Let $I\subseteq\mathbb R$ be an interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $f:I\to\mathbb R$. For $x,y\in I$ and $\lambda\in[0,1]$, the point $\lambda x+(1-\lambda)y$ belongs to $I$. The function $f$ is convex when the convex-combination inequality holds for every weight in $[0,1]$:

$f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y).$

It is strictly convex when this inequality is strict whenever $x\ne y$ and $0<\lambda<1$. It is concave (respectively strictly concave) when $-f$ is convex (respectively strictly convex). It is midpoint convex when the displayed inequality is required only at $\lambda=1/2$.

The endpoint weights $0$ and $1$ impose equalities, so strict convexity does not ask for strictness there. A singleton interval satisfies the convexity and strict-convexity conditions vacuously.