In this recent paper a continuous function with your required property is called pseudo-open. It makes sense, I think, because the property is equivalent to the fact that $f^{-1}(A)$ is an open, dense set whenever $A$ is open and dense.

Notice, however, that in the paper both the domain and the codomain are assumed to be Hausdorff and without isolated points.

In the recent paper Juhász, Soukup, and Szentmiklóssy - Spaces of small cellularity have nowhere constant continuous images of small weight, a continuous function with your required property is called pseudo-open. It makes sense, I think, because the property is equivalent to the fact that $f^{-1}(A)$ is an open, dense set whenever $A$ is open and dense.

Notice, however, that in the paper both the domain and the codomain are assumed to be Hausdorff and without isolated points.

In this recent paper a continuous function with your required property is called pseudo-open. It makes sense, I think, because, as it is proven in the paper, the property is equivalent to the fact that $f^{-1}(A)$ is an open, dense set whenever $A$ is open and dense.

Notice, however, that in the paper both the domain and the codomain are assumed to be Hausdorff and without isolated points.

In this recent paper a continuous function with your required property is called pseudo-open. It makes sense, I think, because the property is equivalent to the fact that $f^{-1}(A)$ is an open, dense set whenever $A$ is open and dense.

Notice, however, that in the paper both the domain and the codomain are assumed to be Hausdorff and without isolated points.