Analog of continuous functions for bornological spaces

Question

Given $(X,T_x,B_x)$ and $(Y,T_y,B_y)$ which are both topological and bornological spaces, a continuous map $f:X\to Y$ is one such that the pre image of open sets is open $f^{-1}(U)\in T_x; U\in T_y$ and an open map is one such that it sends open sets to open sets $f(U)\in T_y; U\in T_x$. There is an analog of open maps in bounded maps $f(U)\in B_y; U\in B_x$, but is there any use for the class of maps that are the analogs of continuous functions $f^{-1}(U)\in B_x; U\in B_y$? I haven’t seen any name for them, but is there a text that delves into them and it’s relations with topological spaces?