Where here we discover some interesting facts about continuous convex functions.
We know that a function is convex if
for all and .
We see that if is a continuous function, then an equivalent condition for convexity is that either of the following inequalities holds
If is convex it is straightforward to prove the right hand side inequality. Conversely, notice that the right-hand side of that inequality is
So then, we have
which means that is convex. Actually, it means that for almost all , but since is assumed to be continuous, it holds for all .
There is another interesting inequality we may prove: We may use the integral form of Jensen’s inequality according to which if is convex and and are integrable on then
Here, choose ; we have
Then, we have the double-sided inequality
Assuming that is continuous, then is convex if and only if at least one of these inequalities holds.