condition of convexity when midpoint convex

Question

Let $f:I \rightarrow R$ be a function satisfying the equation $f(\dfrac{x+y}{2}) \leq \dfrac{f(x)+f(y)}{2}$

The question is,

1)Is $f$ continuous when $I$ is closed?
2)Is $f$ continuous when $I$ is opened?

While searching the internet, there was proof about convexity and continuity, but I can't find about the relation of midpoint convexity and continuous.

Also, is there any intuitive difference about the question (1) and (2)? (I've found some counterexamples about continuity and counterexamples, but none of them showed the proceedings to reach about the counterexample.

For the closed case, what happens when the value of $f$ at an endpoint is very large? — Michael Burr, Apr 17 '19 at 10:44
See also: Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex. (Some of the answers mention also that such example cannot be continuous.) — Martin Sleziak, Apr 29 '19 at 09:44

score 0 · Answer 1 · answered Apr 29 '19 at 08:45

0

A standard (counter)-example are $\mathbb Q$-linear functions on $\mathbb R$. They satisfy the midpoint-convexity inequality but not all of them are continuous.

answered Apr 29 '19 at 08:45

daw

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condition of convexity when midpoint convex

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