0
$\begingroup$

Assume $f:[a,b] \to \mathbb R$ is convex. We want to show $f(a+)=\displaystyle \lim_{x \to a^+} f(x)$ and $f(b-)=\displaystyle \lim_{x \to b^-} f(x)$ exist. I was thinking of first showing that $f$ has right and left derivatives everywhere in $(a,b)$ which are increasing, then use that is $x then slope of line joining $f(x), f(y)$ is greater than $f'_{+}(x)$ and lower than $f'_{+}(y)$ to establish that $f(x)$ is bounded and monotone in some $(a,d)$, $d$ small and use it to derive the existence of right limit to $a$ (similar for $b$).

It seems that this is a bit too complicated though... Is there a simpler way to do it?

  • 0
    In the book by Niculescu, *Convex functions and their applications*, Proposition 1.3.4 solves your problem the same way as you did: by local monotonicity. I am unable to find a more direct approach...2012-09-11

0 Answers 0