TITLE: Example of function satisfying for fixed $t\in (0,1)$ inequality $f(tx+(1-t)y) \leq tf(x)+(1-t)f(y)$
QUESTION [5 upvotes]: I would like to know an example of function $f: \mathbb{R} \rightarrow \mathbb{R}$ which is not convex but satisfies for fixed $t\in (0,1)$ the following inequality:
$$f(tx+(1-t)y) \leq t f(x)+(1-t)f(y) \textrm{  for all  } x,y \in \mathbb{R}.$$
I know only that such functions have to be discontinuous everywhere.

REPLY [2 votes]: Zoltán Daróczy, Zsolt Páles: Convexity with given infinite weight sequences.
Stochastica: revista de matemática pura y aplicada, ISSN 0210-7821, Vol. 11, No. 1, 1987, 5-12, link to pdf. 

Lemma 1. Let $D\subseteq\mathbb R^n$ be a convex compact set. Let $f:D\to\mathbb R$ be an $\alpha$-convex function for some fixed $\alpha\in(0,1)$, i.e. assume that
  $$f(\alpha x+(1-\alpha)y)\le \alpha f(x)+(1-\alpha)f(y) \qquad (\ast)$$
  holds for all $x,y\in D$. Then $f$ is also Jensen convex, i.e. $(\ast)$ is satisfied with $\alpha=\frac12$.

According to the autors, the same result was obtained earlier by N. Kuhn in the paper A note on t-convex functions, but I was not able to find this paper online.

Based on the above result, it suffices to consider $t=\frac12$. It was explained in this answer that some form of AC is needed to construct non-convex Jensen convex function. (Since every measurable Jensen convex function is convex.)