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