Continuity of functions appears throughout the grades in South African high school (FET (further education and training)) topics as prescribed by the final draft of the Curriculum and Assessment Policy Statement. This article reports on the use of a combined framework of APOS (action-process-object-schema) and DCT (dual coding theories) to analyze data captured in a study which investigated second-year teacher trainees' understanding of the concept of continuity. The study is qualitative in that it reports on these teacher trainees' mental constructions of the concept of continuity of single-valued functions, obtained from analysis of their responses to structured activity sheets. The 12 students in this study specialize in the teaching of mathematics for the FET high school curriculum at a South African education faculty. A two-tiered concurrent approach was attempted, one through student-collaborations and the other through an instructional design worksheet, to develop mathematical understandings of the concept of continuity. (Contains 2 tables and 6 figures.)