Concave Real Function is Continuous

From ProofWiki
Jump to navigation Jump to search

Theorem

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


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


Proof

From Concave 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:

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

$\blacksquare$


Also see