Convex Real Function is Continuous

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

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


From Convex Real 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:

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


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