This is just a notation question. 
If I cannot find a preexisting name, I would try one-sided convex, or something of the sort. 



Is there a name for functions f(x,y) that are only convex in x (and continuous in y).
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.


