Measurable functions whose graphs are dense in the plane are well known. Examples include, the Conway 13 function, as given in the answer in this link: When is the graph of a function a dense set?

However, is it possible to impose the extra condition that the (one dimensional) measure of the set formed by taking the projection (on the horizontal axis) of all points on the graph (of our measurable function) that are contained in any given bounded rectangle (with sides parallel to the axes) is strictly positive?

EDIT: Made the question less notational.