| Tartalmi leírás: |
A real valued function (f:Dto mathbb(R)) defined on an open convex subset (D) of a normed space (X) is called emph(rationally ((k,h,d))-convex) if it satisfies
[ fleft(k(t)x + k(1-t)y right) leq h(t) f(x) + h(1-t) f(y) + d(x,y) ]
for all (x,yin D) and (tin mathbb(Q) cap [0,1]), where (d:X times X to mathbb(R)) and (k, h:[0,1] to mathbb(R)) are given functions.
Our main result is of a Bernstein-Doetsch type. Namely, we prove that (under some natural assumptions) if $f$ is locally bounded from above at a point of (D) and rationally ((k,h,d))-convex then it is continuous and ((k,h,d))-convex.
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