Convex Real Function is Continuous

Theorem
Let $f$ be a real function which is either convex or concave on the open interval $\left({a \,.\,.\, b}\right)$.

Then $f$ is continuous on $\left({a \,.\,.\, b}\right)$.

Proof
From Convex or Concave Function is Left-Hand and Right-Hand Differentiable, we have:
 * $\displaystyle \lim_{h \to 0^-} f \left({x + h}\right) - f \left({x}\right) = \left({\lim_{h \to 0^-} \frac {f \left({x + h}\right) - f \left({x}\right)} {h}}\right) \left({\lim_{h \to 0^-} h}\right) = 0$

and similarly:
 * $\displaystyle \lim_{h \to 0^+} f \left({x + h}\right) - f \left({x}\right) = \left({\lim_{h \to 0^+} \frac {f \left({x + h}\right) - f \left({x}\right)} {h}}\right) \left({\lim_{h \to 0^+} h}\right) = 0$

This applies whether $f$ is either convex or concave.