Concave Real Function is Continuous
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Theorem
Let $f$ be a real function which is concave on the open interval $\openint a b$.
Then $f$ is continuous on $\openint a b$.
Proof
From Concave 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 $\openint a b$.
$\blacksquare$