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 Limits of Convex or Concave Function, we have:

$$\lim_{h \to 0^-} f \left({x + h}\right) - f \left({x}\right) = \left({\lim_{h \to 0^-} \frac {f \left({x + h}\right) - f \left({x}\right)} {h}}\right) \left({\lim_{h \to 0^-} h}\right) = 0$$

and similarly

$$\lim_{h \to 0^+} f \left({x + h}\right) - f \left({x}\right) = \left({\lim_{h \to 0^+} \frac {f \left({x + h}\right) - f \left({x}\right)} {h}}\right) \left({\lim_{h \to 0^+} h}\right) = 0$$

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