The literature emphasizes the interest and need to learn the concept of continuity with understanding, although this meaning it is not consensual nor specified how it may emerge and be developed in the learning process. In this paper we report the results of a study aiming to analyse Brazilian preservice mathematics teachers' understanding of the continuity of a function concept, in a context of a teaching experiment which design is based on a theoretical model for learning with understanding that consider three dimensions: the meanings of continuity concept, the use of its different representations and their application in problem solving that involves it. Data collection includes participant observation with audio and video recording of the lessons, students' written and digital work on the tasks proposed during the teaching experiment, and interviews applied to the students. The qualitative and interpretative data analysis showed that students, in general, assigned different meanings to the continuity, which emerged from their concept image. At the end of the teaching experiment, they show an adequate conception of this concept, being able to articulate different representations and perform treatments and conversions to interpret and represent the (dis)continuity of a function, and to correctly apply the criteria of existence of the concept of continuity and the intermediate value theorem in the analysis of errors and in the proving of mathematical propositions, and in solving problems that call for mathematical modelling. Thus, as they revealed a learning with understanding of the continuity concept, the study also provides information to propose innovative mathematics teaching and learning methodologies aimed at improving that learning.
