A few months back, I wrote a post about some elementary results concerning convex functions on the interval. One of these results was that a real-valued continuous function defined on an interval is convex if and only if it is midpoint convex. I want to present another proof of this result, one which comes from Constantin Niculescu’s and Lars-Erik Persson’s text Convex Functions and Their Applications.

**Proposition 1. **Suppose is continuous. Then is convex if and only if is midpoint convex.

*Proof. *Necessity is obvious. For sufficiency, suppose that is not convex; that is, there exist and such that

Define a function by

Note that is the sum of and an affine function. I claim that . Indeed,

Observe that and is midpoint convex, since

Set . Since is continuous, , where the positivity implies that . For any such that , we have that and , so by midpoint convexity of ,

,

which is a contradiction.

I claim that any function satisfying

,

for all and such that , is midpoint convex. Indeed, Fix and set . Then

,

so that

Thus, we obtain a corollary.

**Corollary 2.** A continuous function is convex if and only if

,

for all and such that .

In fact, for Proposition 1, we can relax the sufficiency condition to

for some fixed parameter . Indeed, it is easy to verify that satisfies the same condition as . I claim that there exist , where without loss of generality, , such that

Indeed,

and

So we can make an appropriate choice of satisfying , which uniquely determines . Then and so that

,

which is a contradiction.

We also can relax the hypothesis that is continuous to is bounded from above on compact subintervals of .

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