alternative definition of concave functions Let $g(x): \mathbb{R}\to \mathbb{R}^+$ be a measurable function. Suppose $\{x: g(x)>0\}=(a, b)$. If $$
g(x)-g(x-\alpha) \geq g(y)-g(y-\alpha),\quad x<y, \alpha\geq 0
$$
Can I say $g(.)$ is concave? How can show this?
 A: Choosing $\alpha = y - x$ shows that your condition implies Jensen's inequality
$$ g\left(\frac{x+y}2\right) \le \frac{g(x)+g(y)}2$$
for all $x,y$ - this is known as Jensen convexity. Iterating this we get $t$-convexity $$g\left((1-t)x + ty\right) \le (1-t)g(x) + tg(y)$$ for all rational $t \in [0,1]$. (Maybe just convince yourself that this is true for dyadic rationals, which is a little easier.)
If $g$ is additionally continuous, then the set $$ T = \{ t \in [0,1] \mid (\forall x,y) \; g\left((1-t)x + ty\right) \le (1-t)g(x) + tg(y) \}$$ can be written as an intersection of preimages of closed sets by continuous functions, so it is closed and thus must be all of $[0,1]$.
It turns out measurability and Jensen-convexity imply continuity and thus full convexity: see Theorem II of Blumberg - On Convex Functions.
A: So long as one one is not dealing with closed intervals (as it appears one is not I think there can can be issues for rational convexity (rather than dyadic rational convexity). If it implies convexity and concavity things are completely different.
Or at least continuity at the end pts even for measurable concave function. It applies to the open interval. I am not sure what your condition is; it might be a little stronger such as having increasing increments or some such. So
particularly even if strictly monotonic and the $F(0)=0$ and $F(1)=1$,$F:[0,1]\to [0,1]$ then one only has to deal with continuity from above for $F(0)=0$ concavity, monotonicity and $F(1)=1$ takes care of the right hand end point. See Kuk. The opposite way around for mid-convex functions. 
I think that if its wright convex/schur convex symmetric and midpoint convex (it has an increasing increments conditions), Then issues may not be a worry under strict monotonicity and $F(0)=0$ $F(1)=1$. Presumably there is condition for wright concavity as well.
Kuczma, Marek, An introduction to the theory of functional equations and inequalities. Cauchy's equation and Jensen's inequality. Edited by Attila Gil\'anyi, Basel: Birkh\"auser (ISBN 978-3-7643-8748-8/pbk). xiv, 595~p. (2009). ZBL1221.39041.
