Our purpose in this paper is to investigate and show how pre-service mathematics teachers think about the continuity and differentiability of functions given in the form of both graphical and symbolic  representations and how they link these two basic and important calculus concepts. The study was conducted in two phases. First, the participants were asked to complete several tasks involving determination of whether a function is continuous or not, at given points and the differentiability of the function at those points. Second, in light of the collected data, clinical interviews were done with several students in order to better understand their thinking and reasoning. Analysis of both quantitative and qualitative data revealed some junctions in the findings of this study. From these two sources of data, we were able to construct a picture of the students’ conceptual link between continuity and differentiability of a function. The results confirmed that students demonstrated difficulties determining the continuity and differentiability of a functions at given points and making connections between limit, continuity and differentiability both in symbolic and graphical representations. 

Keywords: Continuity, differentiability, limit, function, misconception, misunderstanding