This is just a notation question. If I cannot find a preexisting name, I would try one-sided convex, or something of the sort.
closed as off-topic by Pietro Majer, Andrey Rekalo, Stefan Kohl, Alexandre Eremenko, Ricardo Andrade Apr 13 at 0:13
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I have not heard the name of such the function (I deal with convex functions)
A function which is continuous wrt. x and wrt. y is called separately convex. Maybe something in this direction.
A function continuous wrt. x and measureble wrt. y is called Caratheodory function. This is another hint.