This study investigates prospective elementary and secondary school mathematics teachers' ways of reasoning about differentiability at a point and corner points while working on a mathematical modelling activity. Adopting a multiple-case study design, the participants of the study were 68 prospective elementary school mathematics teachers enrolled in the 'Calculus-1' course (Case-1) and 20 prospective secondary school mathematics teachers enrolled in the 'Mathematics Teaching Methods' course (Case-2). We adapted a task from the well-known filling bottle problem and implemented it as a regular part of each course. Written group solutions, individual reflection papers and task-based group interviews were the data sources. Constant comparative analysis method was employed in analysing the data. The results from both cases indicated that prospective mathematics teachers frequently relied on roughly drawn graphs while deciding differentiability. They had difficulties in writing an algebraic model and comprehending the meaning of a corner point in the filling water context. Prospective mathematics teachers got confused when they got different results for differentiability at a critical point through the roughly sketched graph and algebraic model. The data also showed that thinking about these concepts while working on such dynamic situations fostered prospective mathematics teachers in developing more productive ways of thinking.