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.



Name of a generalized version of semi-continuity
This property has been studied and it goes by the name [**somewhat continuous**](http://www.google.com/search?q=%22somewhat+continuous%22+topology).


FYI, I was led to this discovery by its mention in the middle of p. 92 of Zbigniew Piotrowski's 1987 survey paper [*A survey of results concerning generalized continuity on topological spaces*](http://web.ysu.edu/gen/stem_generated_bin/documents/basic_module/A_survey_of_results_concerning_generalized_continuity_on_topological_spaces.pdf), and the google search I give above confirmed that this property has been studied a bit.


