A comment to my post Convexity, Continuity, and Jensen’s Inequality asks how we know that , as , in the proof of Lemma 3. It’s actually false that the sequence converges to . What I believe to be a correct proof follows below. I will replace the false proof in the original post soon.

**Lemma 3. **If is midpoint convex and has a point of discontinuity , then .

*Proof. * is discontinuous at , so there exists a such that

Let be given. For such an , define (i.e. the reflection of by ). By midpoint convexity,

which implies that . I claim that there is such that . Observe that if , then the estimate obtained above implies that . But

which proves the claim. Hence, there exists a point such that .

Suppose that we have constructed as desired. Set . Note that . By midpoint convexity,

which implies that

By induction, we obtain a sequence with the property that and .

I claim that every open neighborhood of contains a point such that , for any . Indeed, let a neighborhood and quantity be given. Let be a sufficiently large integer so that . Run the argument above with to obtain a sequence . Observe that

and

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