Maximal operators are a central object in harmonic analysis, The oscillatory behavior of such objects has been an object of study for many authors over the last decades. However, even in the one dimensional case, there are still interesting questions that remain open. In this talk, we will discuss recent developments and open questions about this topic, particularly about the boundedness and continuity for such operators at the derivative level and its relation with some results regarding the regularity improving properties of these operators.