Convex Functions on an Open Interval: Monotone Slopes, Supporting Lines, Continuity, and the Supremum of Rational Supporting Lines

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $I$ be an interval that is an open subset of $(\mathbb{R},d_{\mathbb{R}})$ (Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv)), and let $f:I\to\mathbb{R}$ be convex on $I$.

(i) Increasing slopes. For all $s,t,u\in I$ with $s<t<u$,

$$\frac{f(t)-f(s)}{t-s}\le\frac{f(u)-f(s)}{u-s}\le\frac{f(u)-f(t)}{u-t}.$$

(ii) Infimum of the right difference quotients. For every $x\in I$ the set

$$\Bigl\{\frac{f(y)-f(x)}{y-x}:y\in I,\ y>x\Bigr\}$$

is nonempty and bounded below; its infimum $c_{x}$ is a real number, and

$$\frac{f(x)-f(s)}{x-s}\le c_{x}\le\frac{f(t)-f(x)}{t-x}\qquad\bigl(s,t\in I,\ s<x<t\bigr).$$

(iii) Supporting lines. For all $x,y\in I$,

$$f(y)\ge f(x)+c_{x}\,(y-x).$$

(iv) Continuity. $f$ is continuous on $I$.

(v) Supremum of the rational supporting lines. For every $y\in I$, with the rational numbers read in $\mathbb{R}$ as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (v),

$$f(y)=\sup\bigl\{f(q)+c_{q}\,(y-q):q\in I\cap\mathbb{Q}\bigr\}.$$