presume that midpoint convex F: [0,1] to [0,1]
is monotone increasing (not necessarily strictly ) F(0)=0 ,F(0.5)=0.5 F(1)=1 ,
As its monotone increasing and midpoint convex with F(0)=0 F(1)=1 it will be continuous and convex on [0,1) by lebesgue measurability
However if I note that if midpoint convex F is rationally convex on closed interval one can use F(0.5) =0.5 to show that F(x)\geq x for all rational x>0.5,
ie 0.5=F(0.5)=F(1/3\times 0+2/3\times 0.75)
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