Convex Real Function is Continuous

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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, we have:

$\ds \lim_{h \mathop \to 0^-} \map f {x + h} - \map f x = \paren {\lim_{h \mathop \to 0^-} \frac {\map f {x + h} - \map f x} h} \paren {\lim_{h \mathop \to 0^-} h} = 0$

and similarly:

$\ds \lim_{h \mathop \to 0^+} \map f {x + h} - \map f x = \paren {\lim_{h \mathop \to 0^+} \frac {\map f {x + h} - \map f x} h} \paren {\lim_{h \mathop \to 0^+} h} = 0$

$\blacksquare$


Also see


Sources