Convex Real Function is Continuous

Theorem
Let $f$ be a real function which is convex on the open interval $\openint a b$.

Then $f$ is continuous on $\openint a b$.

Proof
From Convex Real Function is Left-Hand and Right-Hand Differentiable, $f$ is left-hand and right-hand differentiable on $\openint a b$.

From Left-Hand and Right-Hand Differentiable Function is Continuous, $f$ is continuous on $\openint a b$.

Also see

 * Concave Real Function is Continuous