Convex or Concave Function is Continuous

Theorem

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

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

Proof

From Limits of Convex or Concave Function, 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:

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

This applies whether $f$ is either convex or concave.

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 12.16$