I have recently made use of the following generalization of a continuous function, which seems simple enough it ought to have been used before, but I cannot find any references.

We will say a function $f$ has a semi-continuity property if $f^{-1}(U)$ contains an open set whenever $U$ is an open set.

Is this a studied property? If $f:X\to Y$ and $Y$ is disconnected, then this disagrees with usual continuity but does provide some nice properties. For instance, if $D$ is a dense set, then $f(D)$ is dense.

I have recently made use of the following generalization of a continuous function, which seems simple enough it ought to have been used before, but I cannot find any references.

We will say a function $f$ has a semi-continuity property if $f^{-1}(U)$ contains a non-empty open set whenever $U$ is a non-empty open set.

Is this a studied property? If $f:X\to Y$ and $Y$ is disconnected, then this disagrees with usual continuity but does provide some nice properties. For instance, if $D$ is a dense set, then $f(D)$ is dense.